CWP-461
May 2003
Time-Lapse Monitoring with Coda Wave
Interferometry
Alexandre A. Grêt
— Doctoral Thesis —
Geophysics
Defended on July 19, 2004
Committee Chair:
Cp-Advisors:
Committee members:
Prof.
Prof.
Prof.
Prof.
Prof.
Prof.
Ugur Ozbay
John A. Scales
Roelof K. Snieder
Michael L. Batzle
Thomas M. Boyd
Luis Tenorio
Center for Wave Phenomena
Colorado School of Mines
Golden, Colorado 80401
(1) 303 273-3557
Abstract
The coda of seismic waves consists of that part of the signal after the directly
arriving phases. In a finite medium, or in one that is strongly heterogeneous, the coda
is dominated by waves which have repeatedly sampled the medium. Small changes
in a medium which may have no detectable influence on the first arrivals are amplified by this repeated sampling and may thus be detectable in the coda. We refer to
this use of multiple-sampling coda waveforms as coda wave interferometry. We have
exploited ultrasonic coda waves to monitor time-varying rock-properties in a laboratory environment. We have studied the dependence of velocity on uni-axial stress in
Berea sandstone, the temperature dependence of velocity in granite and in aluminum,
and the change in velocity due to an increase of water saturation in sandstone. Furthermore, We applied coda wave interferometry to seismic data excited by a hammer
source, collected at an experimental hard rock mine, Idaho Springs, CO. We carried
out a controlled stress-change experiment in a pillar and we were able to monitor the
internal stress change. We used coda wave interferometry to monitor temporal changes
in the subsurface of the Mt. Erebus Volcano, Antarctica. Mt. Erebus is one of the
few volcanoes known to have a convecting lava lake. The convection provides a repeating seismic source producing seismic energy that propagates through the strongly
scattering geology in the volcano. There are many other possible applications of coda
wave interferometry in geophysics, including dam and nuclear waste deposit monitoring, time-lapse reservoir characterization, earthquake relocation, stress monitoring in
surface mining and rock physics.
i
To Adrienne
ii
“Dear Mr. Freud
...The quest of international security involves the unconditional surrender by every
nation, in a certain measure, of its liberty of action–its sovereignty that is to say–and it is
clear beyond all doubt that no other road can lead to such security...
Yours very sincerely,
A. Einstein”
“Dear Mr. Einstein
...There is but one sure way of ending war and that is the establishment, by common
consent, of a central control which shall have the last word in every conflict of interests...
Yours,
Sigm. Freud”
Why War?
The Einstein-Freud correspondence (1931-1932)
iii
iv
Table of Contents
vii
Acknowledgments
1
Chapter 1 Introduction
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
What are coda waves? . . . . . . . . . . . . . . . . . . . . . . . .
Reproducibility of coda waves . . . . . . . . . . . . . . . . . . . .
De-correlation of coda waves by changes in the medium . . . . .
Cross-correlation function, the heart of coda wave interferometry
Velocity changes . . . . . . . . . . . . . . . . . . . . . . . . . . .
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Similar methods . . . . . . . . . . . . . . . . . . . . . . . . . . .
The coda wave interferometry bunch . . . . . . . . . . . . . . . .
Data contributions . . . . . . . . . . . . . . . . . . . . . . . . . .
A collection of papers . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 2 Coda Wave Interferometry for Estimating Nonlinear Behavior in
Seismic Velocity
17
2.1
2.2
2.3
2.4
2.5
Abstract . . . . . . . . . .
Introduction . . . . . . . .
Coda wave interferometry
Laboratory experiment . .
Acknowledgments . . . . .
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17
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Chapter 3 Time-lapse Monitoring of Rock Properties with Coda Wave Interferometry
27
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Sensitivity of coda waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Estimation of velocity change from coda waves . . . . . . . . . . . . . . . .
Monitoring uni-axial stress in Berea sandstone . . . . . . . . . . . . . . . .
Monitoring water saturation in Berea sandstone . . . . . . . . . . . . . . . .
Monitoring thermally induced velocity changes in aluminum . . . . . . . . .
Monitoring thermally induced velocity change and acoustic emissions in granite
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
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27
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49
Chapter 4 Monitoring stress change in an underground mining environment
with coda wave interferometry
53
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
Abstract . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . .
Short introduction to coda wave interferometry
Experimental setup and local geology . . . . .
Reproducible seismic waveforms . . . . . . . . .
Decorrelation of coda waves with stress change
Monitoring stress change . . . . . . . . . . . . .
Discussion . . . . . . . . . . . . . . . . . . . . .
Conclusions . . . . . . . . . . . . . . . . . . . .
Acknowledgments . . . . . . . . . . . . . . . . .
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53
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Chapter 5 Monitoring rapid temporal change in a Volcano with Coda Wave
Interferometry
67
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . .
Mount Erebus, it’s eruptions and the seismic network
Reproducible seismic events at Mount Erebus . . . . .
Decorrelation of coda waves . . . . . . . . . . . . . . .
Measurements at different locations . . . . . . . . . . .
Conclusions . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgments . . . . . . . . . . . . . . . . . . . . .
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79
Chapter 6 Time versus frequency resolution of small changes
6.1
6.2
6.3
6.4
6.5
Abstract . . . . . . . . . . . . . . . . . . . . . . .
Overview . . . . . . . . . . . . . . . . . . . . . .
The Fabry-Perot etalon . . . . . . . . . . . . . .
6.3.1 Time versus frequency in the ideal case .
6.3.2 Thinking in frequency from the beginning
A canonical experiment . . . . . . . . . . . . . .
6.4.1 Time domain . . . . . . . . . . . . . . . .
6.4.2 Frequency domain . . . . . . . . . . . . .
6.4.3 Time-frequency domain . . . . . . . . . .
6.4.4 Comparison of the different methods . . .
Acknowledgments . . . . . . . . . . . . . . . . . .
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References
vi
Acknowledgments
I would like to thank Prof. Roel Snieder who started the whole coda wave interferometry avalanche. I am grateful for all his great ideas and all the exciting discussions on
scattering, coda, volcanoes, cycling and an endless list of scientific, political and personal
topics. The door of Roel’s office was always open and he always had an open ear for my
questions and problems, especially during times when I wanted to ditch the whole Ph.D.
stuff.
My very special gratitude goes to Prof. John A. Scales. I have rarely seen a person
with more enthusiasm and energy. John not only created the Physical Acoustics Laboratory
but by doing so he built a place where graduate and undergraduate students, professors,
visitors, family and Charlie meet to talk about anything and everything. If I’d have to pick
the single most valuable thing I learned from John it would be his attitude toward research,
or as he likes to say “let’s do it!”
It would be meaningless to rave about the Lab without acknowledging Kasper van
Wijk, Albena Mateeva, Alison Malcolm, Brian Zadler, Hans Ecke and Matt Haney who all
carry the vibes of this special place.
Without Prof. Mike Batzle none of my experimental work would have been possible. Every experiment starts with consulting Mike and the Center for Rock Physics where
anything is possible, even creating our own beer label.
The Center for Wave Phenomena and its sponsors supported me and not only financially. In particular, I thank Prof. Ken Larner for making sure that the Center for
Wave Phenomena can continue to be a learning place for all of us. I acknowledge Barbara
McLenon, Michelle Szobody and Sara Summers for making the wheel go round. I extend
my thanks to all the faculty and student members of the Center for Wave Phenomena and
the Department of Geophysics, for their friendship and encouragements.
I thank Prof. Luis Tenorio who is not only a mathematician asking awkward questions
but a friend who has partial responsibility for my caffeine addiction. Luis treated me more
as a colleague than a student and that is extremely encouraging, especially at times when
nothing seems to work.
I owe it all to my beloved wife Adrienne. She joined me for this adventure, thousands
of miles away from family and old friends. Adrienne, thank you for being with me during
all the difficult and all the easy times. Matéo, I had no idea how much energy can come
from your smiles!
vii
Alexandre A. Grêt
/
Time-Lapse Monitoring with Coda Wave Interferometry
1
Chapter 1
Introduction
In this introduction, I give a birds-eye view of my work for the non-specialist. I
highlight the important ideas and concepts, provide an intuitive explanation of the physics,
leaving references, equations and details to the following chapters. I hope that most readers
will be excited enough to read on to the end of the thesis.
1.1
What are coda waves?
The first waves we generally record after an earthquake are P-waves (primary) followed
by S-waves (secondary) (Figure 1.1). As these body waves interact with the surface, some
of their energy is transformed into surface waves. Surface waves have large amplitudes
and reach the seismometer after the first arriving P and S waves. After that, the recorded
amplitudes decrease steadily. This decaying wave-train after an earthquake is composed of
a mixture of P, S and surface waves that arrive along scattered paths through the complex
structure of the earth. It is this part of a seismogram that is called the coda. This word is
used by analogy with the final section of a musical composition (Latin cauda, tail).
Although the word coda waves originated in the earthquake seismology community,
its use does not have to be restricted to seismograph readings. Measurements from exploration seismology and ultrasonic laboratory experiments generally contain coda energy.
Examples of ultrasonic coda measurements are given in Chapters 2 and 3, while Chapter 4
contains coda wave data from a hammer source seismic experiment and Chapter 5 includes
earthquake data recorded at a volcano.
While we usually have a good understanding of the traveled paths and velocities of the
first arriving P and S waves, our knowledge about coda waves is more limited. Unraveling
the different wave modes and propagation paths forming the coda is in most cases not
possible. We often can, however, explain part of it. A P-wave for example can bounce
back and forth between two boundaries many times (Figure 1.2a). The P wave arrives at
the receiver in constant time intervals, leaving a distinct pattern of wave arrivals in the
coda measurement. I like to think of this as a bouncing ball mode. A second example is a
phenomenon often seen in global seismology, where surface waves created by an earthquake
circle the earth many times (Figure 1.2b), which leaves a distinct pattern in the coda
energy as well. A more complex behavior is the multiple reflection from boundaries where
the waves reflect in all directions in random fashion (Figure 1.2c), or the scattering from
inhomogeneities within the medium of propagation (Figure 1.2d). In the last two examples,
it is difficult to recognize a pattern in the coda measurement.
2
Chapter 1. Introduction
P−wave
surface wave
S−wave
Figure 1.1. This seismogram shows the vertical motion of the earth in Kevo, Finland. The
record begins at 500 seconds (8.3 minutes) after the earthquake occurred in Loma Prieta,
California, near San Francisco. The first arrivals for P-, S- and surface waves are indicated
by arrows.
Alexandre A. Grêt
/
Time-Lapse Monitoring with Coda Wave Interferometry
a
b
c
d
3
Figure 1.2. Cartoon of different wave modes that may co-exist in a medium. a) A bouncing
ball mode where the waves go straight back and forth between two boundaries. b) Surface
waves that propagate along the boundary, circling the medium. c) More complex reverberations between the boundaries. d) Multiple scattering from small-scale scatterers (small
circles) in the medium.
4
Chapter 1. Introduction
a
10
b
15
20
time (s)
25
30
35
Figure 1.3. Two waveforms recorded at the Merapi volcano. The blue and the black lines
represent two measurements taken at different days, two weeks apart in time. The two bars
(a and b) indicate the position and duration of the time window shown in Figures 1.4 and
1.5.
In contrast to a wave that propagates in a direct path from source to receiver, coda
waves travel along much longer and more complex trajectories. This long propagation path
of coda waves is the key observation in order to understand Coda Wave Interferometry.
A large part of chapter 6 is devoted to the discussion of path length and the increased
sensitivity of coda waves due to long paths.
1.2
Reproducibility of coda waves
Coda Waves may have a noise appearance and are therefore often discarded by geophysicists in their analysis. They are, however, very different from random noise. In fact,
there are many examples that show their high reproducibility. Figure 1.3 shows two seismic
waveforms, recorded by Ulrich Wegler and colleagues from the University of Podsdam. The
seismograms were recorded at the Merapi volcano in Indonesia. A repeatable source (air-
Alexandre A. Grêt
/
Time-Lapse Monitoring with Coda Wave Interferometry
5
a
11
11.5
12
time (s)
12.5
13
Figure 1.4. A short time window of the same data presented in Figure 1.3. For comparison,
the two measurements are plotted on top of each other (blue and black lines represent the
same measurements as in Figure 1.3).
6
Chapter 1. Introduction
b
21.5
22
22.5
23
time (s)
23.5
24
Figure 1.5. A short time window of the coda for the same data presented in Figure 1.3. For
comparison, the two measurements are plotted on top of each other (blue and black lines
represent the same measurements as in Figure 1.3).
Alexandre A. Grêt
/
Time-Lapse Monitoring with Coda Wave Interferometry
7
gun in a water-pool) is fired and the resulting seismic signal is recorded by the seismometer
located at the other side of the volcano. The blue line in Figure 1.3 shows the recorded
waveform. One week later the air-gun is fired again, and the same seismometer, at the same
location, records the resulting waveforms (black line in Figure 1.3). The two waveforms are
almost identical. If we inspect the early part (first arriving waves) of the two measurements
(Figure 1.4), we see the similarity even clearer. Furthermore, Figure 1.5 shows part of the
coda of the two measurements and the waveforms match wiggle by wiggle. In other words,
given a repeatable source at a fixed location, a fixed receiver, and nothing changes in the
medium through which the waves propagate, the recorded signals remain identical, including the coda waves. Chapters 4 and 5 show other good examples of reproducible seismic
coda measurements.
Evidently, seismic data contain random noise, which leads to differences in the measurements. In practice we may need to filter and/or average records to reduce noise. In a
laboratory experiment, where we use ultrasonic waves, we can easily average hundreds or
thousands of measurements within seconds. This results in data that are virtually noise
free. The ultrasonic data used in Chapters 2 and 3 are such examples.
1.3
De-correlation of coda waves by changes in the medium
As long as we keep the source, receiver and medium traversed by the waves unchanged,
we record an identical signal (up to random noise). Figure 1.6 shows the records for the
seismic experiment at the Merapi volcano, discussed in the previous paragraph, but the
time elapsed between the two records is now one year. As expected, the two waveforms
are very similar. Comparing the early arriving waves (Figure 1.7 ), again displays the
reproducibility of the measurement. Comparing the coda shown in Figure 1.8, however,
shows that the waveforms don’t match. Since we demonstrated the repeatability of the
seismic source and the recording equipment in the previous paragraph, this difference in
the coda can only originate from a change in the subsurface. An important observation
is that one coda measurement is almost a time-shifted version of the other. We will later
relate that time-shifts to differences in seismic velocity.
It is important to note that the two measurements are not completely different; their
early part remains largely unchanged while the later arriving waves are altered. In other
words, the change in the medium has little effect on the early arriving waves. These early
arriving waves can therefore not be used to detect the change. The coda waves are, however,
more sensitive to the change and can therefore be used to estimate the change in the medium.
The increased sensitivity of coda waves as compared to that of the early arrivals can be
explained by differences of the length of paths traversed by these waves. For example, a
surface wave that circles the earth five times is five times more sensitive than a wave that
only makes one trip around the globe. Chapter 6 is entirely devoted to the sensitivity
question, both in time- and frequency domain measurements.
8
Chapter 1. Introduction
a
10
b
15
20
25
time (s)
30
35
Figure 1.6. Two waveforms recorded at the Merapi volcano. The blue and the black lines
represent two measurements taken at different days (one year apart). The two bars (a and
b) indicate the position and duration of the time window shown in Figures 1.7 and 1.8.
Alexandre A. Grêt
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Time-Lapse Monitoring with Coda Wave Interferometry
9
a
11
11.5
time (s)
12
12.5
Figure 1.7. A short time window of the same data shown in Figure 1.6. For comparison,
the two measurements are plotted on top of each other (blue and black lines represent the
same measurements as in Figure 1.6).
10
Chapter 1. Introduction
b
21.8
22
22.2
22.4
22.6
time (s)
22.8
23
23.2
Figure 1.8. A short time window of the coda for the same data presented in Figure 1.6. For
comparison, the two measurements are plotted on top of each other (blue and black lines
represent the same measurements as in Figure 1.6).
time lag (s)
Figure 1.9. Cross-correlation function of the data shown in Figure 1.8. Note that the
maximum of the cross-correlation function is centered at δt which is the value we try to
estimate.
Alexandre A. Grêt
1.4
/
Time-Lapse Monitoring with Coda Wave Interferometry
11
Cross-correlation function, the heart of coda wave interferometry
In the previous two paragraphs we have talked about waveforms that are identical,
similar or different. But how do we quantify how similar or different these waves are? The
tool of choice to compute the degree of similarity of two signals is the cross-correlation
function (Figure 1.9).
The maximum of the cross-correlation function (R max ) characterizes how similar two
signals are, where Rmax = 1 means that they are identical. The time at the maximum (δt),
is the time by how much the two signals have to be shifted with respect to each other to
achieve the best possible match. We use R max in Chapters 2 and 5, whereas we use δt in
chapters 2, 3, 4 and 6 to monitor changes with Coda Wave Interferometry.
Note that it is this comparison of two waveforms, where one wave has propagated
through an un-changed and one through a changed medium, which is similar to an interferometer, lending our method its name. Furthermore, an optical interferometer can be
made more sensitive by lengthening the propagation path of the light, the Fabry-Perot
interferometer is an example (6).
As explained in the previous paragraph, the sensitivity increases with path length.
Similarly, there is a sensitivity increase with total travel time (twice around the globe
takes twice as long than going around once). In order to compare waves that have similar
sensitivity, we compare small parts of the whole signal by computing R max and δt for
different time windows.
1.5
Velocity changes
Often we can extract information about changes in a medium by monitoring the velocity of wave propagation. I explain the main idea in the following toy example. Figure 1.2b
is a sketch of a surface wave that circles the medium repeatedly. What we would record at
a fixed point on the boundary, is a series of surface waves arriving in constant time intervals
(Figure 1.10). If we reduce the velocity in the medium and repeat the experiment, we measure a similar series of surface waves but with a lower velocity as it takes the wave longer
to make one round-trip and arrives at the receiver with a time lag δt. The wave that circles
twice has a time lag 2δt, the one that makes three trips 3δt and so forth (Figure 1.10a). We
can estimate the time lag from the data by means of the cross-correlation function in a time
window centered around the arrival of one wave. It is important to note that the velocity
decrease might be too small to be detected by a wave that samples the medium once, but
as time elapses the lag becomes larger and larger (Figure 1.10b). If the propagation path
before and after the velocity change remains fixed, the relative time lag (δt/t) is a direct
measure of the relative velocity change (δv/v). In chapters 2, 3, 4 and 6, I show how we
can use this coda wave interferometry to monitor relative velocity changes, in applications
that range from small rock samples in the laboratory to seismic waves in a gold and silver
mine.
For all the applications of coda wave interferometry described in this thesis, we assume
a homogeneous change in the whole medium. In other words we gain sensitivity at the
12
Chapter 1. Introduction
2
t
(a)
time lag
v
v
(b)
t
total travel time t
Figure 1.10. (a) Sketch of a surface wave that circles the medium repeatedly. The measured
wave-train consists of a series of surface waves arriving in constant time intervals. Waves
before (solid line) and after the velocity change (dashed line). (b) A change in velocity leads
to a shifting of the wave-trains, where the shift becomes larger with increasing time.
expense of spatial resolution. This is certainly, however, not always a good approximation.
Some arguments of how we can check weather or not the change occurs in the entire medium
are given in Chapter 4. Furthermore, Pacheco & Snieder (2003) developed a theoretical
extension to coda wave interferometry that allows the spatial localization of the changed
area, based on multiply scattered waves.
1.6
Applications
As described in the previous paragraphs, we can use coda wave interferometry to
monitor small changes in a medium. In Chapters 2 through 5 we monitor changes at the
laboratory scale up to the scale of monitoring a volcano. In this paragraph I give a preview
of the applications described in the corresponding chapters.
Ultrasonic wave velocity in solids depends on temperature. In most cases, the velocity
decreases with an increase in temperature. We use this in Chapters 2 and 3 where we
introduce a controlled velocity change in aluminum and granite samples. We alter the
temperature of the samples and monitor the associated velocity change with coda wave
interferometry. In contrast to aluminum, where we find a linear dependence of velocity
on temperature, we observe non-linear dependence of velocity on temperature in granite.
This can be attributed to internal cracking that produces acoustic emissions. The cracking
causes an irreversible damage as the granite is heated and cooled.
Alexandre A. Grêt
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Time-Lapse Monitoring with Coda Wave Interferometry
13
In addition to the temperature dependence of ultrasonic velocity, we study the effects
of water saturation and stress change on the velocity in Berea sandstone (Chapter 3). There
are many potential applications for coda wave interferometry in rock physics. Deformation,
phase change of fluids, compaction of granular materials, and steam injection are a few
examples of changes that can be monitored with coda wave interferometry.
Stress change, and deformation are important in mining hazard and safety applications. Inspired by the stress change experiment in the laboratory, we conducted a hammer
source seismic experiment in an underground mine (Chapter 4). We introduced a controlled
internal stress change in a rock pillar with a hydraulic pressure cell introduced into the rock
formation. We monitor internal stress changes in the rock pillar with coda wave interferometry. The method can be used to monitor the rate of stress change and deformation in
the rock mass around excavations, both in underground and surface mines.
Volcanoes are known for their highly complex subsurface, which often results in the
multiple scattering of waves that propagate through the volcano’s underground. At Mt.
Erebus (Antarctica), eruptions of a lava lake located in the volcano’s crater acts as a repeatable seismic source. The earthquakes recorded at a seismic network around the volcano
contain a highly reproducible coda. We use these multiply scattered coda waves to monitor
a rapid temporal change in the subsurface of the volcano (Chapter 5).
Waves that follow a distinct pattern of propagation (Figure 1.2), can be viewed either
as modes or as traveling waves, the modes and the traveling waves are complementary
views of the wave propagation. In resonance spectroscopy such modes are excited and the
frequency response is measured. This is a purely frequency domain measurement that has
similarities with coda wave interferometry, which is a time domain technique. In chapter
6, we discuss the similarities and differences between the two techniques, with a particular
focus on the comparison of the sensitivity to detect changes.
With all the applications of coda wave interferometry included in this thesis, we only
cover the most basic aspects, in the sense that each application could be explored in much
greater depth, and independently be the topic of a Master or Ph.D. thesis. The in-depth
study of velocity change due to temperature on the basis of intra molecular bonding or the
mechanics and physics of crack formation between crystal boundaries in rocks are only two
examples. It is the focus of this thesis to study the feasibility of this technique to a variety
of applications, with the hope that it leads to future research projects.
1.7
Similar methods
Diffusive wave spectroscopy or acoustic diffusive wave spectroscopy in particular, is a
technique similar to coda wave interferometry. In contrast, it uses the diffusion approximation to model the wave-field. We do not make that requirement in coda wave interferometry,
but only assume that the coda wave, sample the medium multiple times.
Cavity ring-down spectroscopy is a technique in optics, which is often used in atmospheric pollution analysis and optical probing of flames. A laser pulse bounces back and
fourth between two mirrors, sampling the medium between the mirrors multiple times (similar to Figure 1.2a). Analogous to coda wave interferometry, the method makes use of the
14
Chapter 1. Introduction
fact that the signal is directly proportional to the length of the path traveled by the laser
light. In cavity ring-down spectroscopy the geometry of the optical cavity (space between
the mirrors) is known and used to compute an absolute velocity or absorption value of
the medium in the cavity. In contrast, coda wave interferometry compares a measurement
before with a measurement after a change, resulting in a relative quantity. This has the
advantage that we do not need any knowledge about the geometry of the propagated paths.
1.8
The coda wave interferometry bunch
Many Researchers have been working on coda wave interferometry projects, which
greatly inspired, influenced and helped my work. In this section I give a short list of the
different projects and at the same time I want to thank the whole group for helping me
with my research.
Roel Snieder and Huub Douma laid the foundation with the theory and numerical
simulations. They showed how we can extract information about the change in scatterer
location from coda waves (Snieder & Douma, 2000). Furthermore, Huub Douma worked on
the change in scatterer strength and Roel Snieder developed an extension for elastic waves
(Snieder, 2002) and used coda wave interferometry to monitor changes in volcanic interiors
(Snieder & Hagerty, 2004). Carlos Pacheco’s masters thesis shows how we can detect a
localized change with coda wave interferometry (Pacheco, 2003) and Mark Vrijlandt used
coda wave interferometry to constrain the relative source location of earthquake doublets
(Snieder & Vrijlandt, 2004).
1.9
Data contributions
The ultrasonic data I used to monitor uni-axial stress on Berea sandstone (chapter 3)
where recorded by Debashish Sarkar and Robert Kranz (Sarkar et al., 2003). Rick Aster
from New Mexico Tech, Socorro, NM, provided the seismic data, that we use to monitor
rapid changes in the subsurface of the Mt. Erebus Volcano (chapter 5).
1.10
A collection of papers
Chapters 2 through 6 are published or submitted to scientific peer-reviewed journals.
• Chapter 2: Coda Wave Interferometry for Estimating Nonlinear Behovior in Seismic
Velocity, Science 2002, by Roel Snieder, Alexandre Grêt, Huub Douma and John
Scales.
• Chapter 3: Time-Lapse Monitoring of Rock Properties with Coda Wave Interferometry, submitted to the Journal of Geophysical Research 2004, by Alexandre Grêt, Roel
Snieder and John Scales.
• Chapter 4: Monitoring Stress Change in an Underground Mining Environment with
Coda Wave Interferometry, submitted to Mining Engineering 2004, by Alexandre Grêt,
Roel Snieder and Ugur Ozbay.
Alexandre A. Grêt
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Time-Lapse Monitoring with Coda Wave Interferometry
15
• Chapter 5: Monitoring Rapid Temporal Changes in a Volcano with Coda Wave Interferometry, submitted to Geophysical Research Letters 2004, by Alexandre Grêt, Roel
Snieder and Rick Aster.
• Chapter 6: Time-Domain versus Frequency-Domain Resolution of Small Changes,
submitted to American Journal of Physics 2004, by Brian Zadler, Alexandre Grêt and
John Scales.
16
Chapter 1. Introduction
Alexandre A. Grêt
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Time-Lapse Monitoring with Coda Wave Interferometry
17
Chapter 2
Coda Wave Interferometry for Estimating
Nonlinear Behavior in Seismic Velocity
2.1
Abstract
In coda wave interferometry one records multiply scattered waves, at a limited number
of receivers, to infer changes in the medium over time. With this technique we have determined the nonlinear dependence of the seismic velocity in granite on temperature and the
associated acoustic emissions. This technique can be used in warning mode, to detect the
presence of temporal changes in the medium, or in diagnostic mode, where the temporal
change in the medium is quantified.
2.2
Introduction
In many applications, such as non-destructive testing, monitoring of volcanoes or radioactive waste disposal sites, one is primarily interested in detecting temporal changes in
the structure of the medium. Temporal changes in Earth’s structure accompanying earthquakes have been observed based on the attenuation of coda waves (Chouet, 1979; Aki, 1985;
Sato, 1986; Got et al., 1990), on the arrival times of the directly arriving waves (Leary et al.,
1979), on velocity changes inferred from later arriving waves (Poupinet et al., 1984; Ratdomopurbo & Poupinet, 1995; Dodge & Beroza, 1997; Li et al., 2000; Snieder & Douma,
2000) (see also (Roberts et al., 1992)), and on changes in seismic anisotropy (Bokelmann &
Silver, 2002; Miller & Savage, 2001). Here we introduce coda wave interferometry whereby
multiply scattered waves are used to detect temporal changes in a medium by using the
scattering medium as an interferometer. For quasi-random perturbations of the positions
of point scatterers, or for a change in the source location, or for a change in the wave
velocity, estimates of this perturbation can be derived from multiply scattered waves by a
cross-correlation in the time domain.
In the numerical example (Figure 2.2), the wave-field for a medium consisting of
isotropic point scatterers is computed using a deterministic variant (Groenenboom & Snieder,
1995; Snieder & Scales, 1998) of Foldy’s method (Foldy, 1945). Given the mean free path
(l = 20.1 m) and the wave velocity (v = 1, 500 m/s) one can infer that after t = 5.4 × 10 −2
s the waves are on average scattered more than three times. The later part of the signal
is called the coda. Suppose that one repeats this multiple scattering experiment after the
scatterer locations are perturbed. The perturbation in the scatterer location 1/30 of the
18Chapter 2. Coda Wave Interferometry for Estimating Nonlinear Behavior in Seismic Velocity
∗
Strong multiple scattering
0.02
0.04
0.06
t (s)
0.08
0.10
Figure 2.1. Location of 100 scatterers before and after the perturbation (filled dots and
open dots respectively) with the source (asterisk) and receiver location (triangle). For the
sake of clarity the scatterer displacement is exaggerated with a factor 40. The scatterers are
placed in an area of 40 × 80 m. The waveforms recorded before and after the perturbation
at the receiver are shown on the right with a solid and dashed line respectively.
dominant wavelength and is uncorrelated between scatterers (Maret & Wolf, 1987; Heckmeier & Maret, 1997).
In this example the scatterers locations are perturbed. In general, a perturbation
can involve other changes in the medium or a change in source location. We refer to
the waveform before the perturbation as the unperturbed signal, and to the waveform
after the perturbation as the perturbed signal. For early times (t < 0.04 s) the waves in
Figure 2.2 have not scattered often, rendering the path lengths of these waves insensitive
to the small perturbations of the scatterers (small compared to the dominant wavelength
λ = 2.5 m), which causes the unperturbed and perturbed signals to be similar. However,
the multiply scattered waves are increasingly sensitive with time to the perturbations of the
scatterer locations because the waves bounce more often among scatterers as time increases.
The correlation between the unperturbed and perturbed signals therefore decreases with
increasing time.
2.3
Coda wave interferometry
The perturbation in the medium can be retrieved from the cross-correlation of the
coda waves recorded before and after the perturbation. The unperturbed wave-field can be
Alexandre A. Grêt
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Time-Lapse Monitoring with Coda Wave Interferometry
19
δ (m)
0.125
0.100
0.075
0.050
0.025
0.10
0.15
t (s)
0.20
0.25
Figure 2.2. The value of δ obtained from the time-windowed cross correlation of the wave
forms in Figure 2.2 and of 20 other receivers as a function of the center time t of the time
window (solid line) plus or minus one standard deviation (dotted lines) for T = 2 × 10 −2 s.
The true root mean square displacement value δ true = 8 × 10−2 is shown by the horizontal
solid line.
20Chapter 2. Coda Wave Interferometry for Estimating Nonlinear Behavior in Seismic Velocity
written as a Feynman path summation (Snieder, 1999) over all possible paths P :
X
uunp (t) =
AP S(t − tP ) ,
(2.1)
P
where a path is defined as a sequence of scatterers that is encountered, t P is the travel time
along path P , AP is the corresponding amplitude and S(t) is the source wavelet. When the
perturbation of the scatterer locations (or source location) is much smaller than the mean
free path, the effect of this perturbation on the geometrical spreading and the scattering
strength can be ignored and the dominant effect on the waveform arises from the change in
the travel time τP of the wave that travels along each path:
X
uper (t) =
AP S(t − tP − τP ) .
(2.2)
P
The time-windowed correlation coefficient is computed from:
R
(t,T )
R t+T
0
0
0
t−T uunp (t )uper (t + ts )dt
(ts ) ≡ R
1/2 ,
R t+T
t+T 2
0
0
2
0
0
t−T uunp (t )dt t−T uper (t )dt
(2.3)
where the time window is centered at time t with duration 2T and t s is the time shiftPused
in the cross-correlation. When equations (2.1) and (2.2) are inserted, double sums P P 0
over all paths appear. In these double sums, the cross-terms with different paths (P 6= P 0 )
are incoherent and average out to zero when the mean of the source signal vanishes. This
means that in this approximation:
P
2
P (t,T ) Ap C(τP − ts )
(t,T )
P
,
(2.4)
R
(ts ) ≈
2
P (t,T ) Ap C(0)
P
where P (t,T ) denotes a sum over the paths with arrival times within the time window
of
and the auto-correlation of the source signal is defined as C(t) ≡
R ∞the cross-correlation,
0 + t)S(t0 )dt0 .
S(t
−∞
For time shifts τ much smaller than
the dominant period, a second-order Taylor expansion gives C(τ ) = C(0) 1 − 12 ω̄ 2 τ 2 , where ω̄ 2 is the mean-squared frequency of the
multiply scattered waves that arrive in the time window. Using this gives
1 R(t,T ) (ts ) = 1 − ω̄ 2 (τ − ts )2 (t,T ) ,
2
(2.5)
where h· · · i(t,T ) denotes the average for the wave paths with arrivals in the time interval
(t − T, t + T ).
The time shifted cross correlation R (t,T ) (ts ) has a maximum when
ts = hτ i(t,T ) ,
(2.6)
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Time-Lapse Monitoring with Coda Wave Interferometry
21
where hτ i(t,T ) is the mean travel time perturbation of the arrivals in the time window. The
value of the cross correlation at its maximum is given by
1
(t,T )
Rmax
= 1 − ω̄ 2 στ2 ,
2
(2.7)
with στ2 the variance of the travel time perturbations for waves arriving within the time
window. This means that the mean and the variance of the travel time perturbation of the
waves arriving in the employed time window can be extracted from the data recorded with
a repeatable source, and one or more receivers.
Different types of perturbations leave a different imprint on the time shifted correlation
coefficient. When the scatterer locations are perturbed independently with root mean square
displacement δ, the mean travel time perturbation vanishes (hτ i (t,T ) = 0) and the variance
is given by (Snieder, 1999)
2δ 2 t
,
(2.8)
στ2 =
vl∗
where l∗ is the transport mean free path (Maret, 1995; Lagendijk & van Tiggelen, 1996).
In deriving (2.8) we assume that the number of scatterers encountered is on average given
by n = vt/l, where t is the time that the wave has spent in the scattering medium. Using
(2.7) and (2.8), the root mean square perturbation of the scatterer location follows from
the maximum of the time-windowed correlation coefficient:
(t,T ) vl∗
δ 2 = 1 − Rmax
.
(2.9)
ω̄ 2 t
A different type of perturbation is a constant change δv in the velocity for fixed locations of the scatterers. The mean travel time perturbation is given by hτ i (t,T ) = −(δv/v)t,
and when the time window is small (T t), σ τ = 0. The velocity change follows from the
time of the maximum of the time-shifted cross correlation function:
δv
tmax
=−
.
v
t
(2.10)
When the perturbation consists of a displacement of the source location over a distance
δ for a fixed medium, only the wave-path to the first scatterer is perturbed. In that case the
mean travel time perturbation vanishes hτ i (t,T ) = 0 and for an isotropic source the variance
is given by στ2 = (δ/v)2 . The source displacement then follows from
(t,T )
).
δ 2 = 2v 2 /ω̄ 2 (1 − Rmax
(2.11)
These different perturbations can be distinguished on the basis of the time shifted
cross correlation. When the positions of the scatterers are perturbed, the mean travel time
perturbation vanishes and the maximum of the cross correlation decreases linearly with
increasing time, whereas for the perturbation of the source position the maximum value of
this function is independent of time. A change in the velocity is detectable by a shift in the
position of the maximum of R (t,T ) (ts ) that increases linearly with time.
22Chapter 2. Coda Wave Interferometry for Estimating Nonlinear Behavior in Seismic Velocity
1.5
1.0
0.0
−1.0
0.5
0.01
0.02
0.03
0.04
1.53
1.54
1.55
0.1
−0.5
0.0
−0.1
1.52
−1.5
0.0
0.5
1.0
Time (ms)
1.5
2.0
Figure 2.3. Wave-forms recorded in the granite sample for temperatures of 45 ◦ C (in blue)
and 50◦ C (in red) respectively. The insets show details of the wave-forms around the first
arrival (top) and in the late coda (bottom).
The root mean square displacement of the scatterers inferred from the numerical example (Figure 2.2) is shown in Figure 2.3 as a function of the center time t of the time
window. The inferred change δ in the scatterer location does not depend on the center
time of the window used for the cross correlation. This provides a consistency check of the
method.
2.4
Laboratory experiment
The extreme sensitivity of the coda waves to changes in the medium is used here in
a laboratory experiment to infer the temperature dependence of the seismic velocity in
Elberton granite. In many experiments, the change in the seismic velocity in rock samples
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Time-Lapse Monitoring with Coda Wave Interferometry
23
0.004
4000
0.003
3000
0.002
2000
0.001
1000
0.000
Counts
| dv/v |
Alexandre A. Grêt
20
40
60
80
Temperature (degree Celsius)
0
100
Figure 2.4. The absolute value of the relative velocity change for a 5 ◦ C increase (red symbols) and 5◦ C decrease (blue symbols) respectively as a function of the highest temperature
during the change. The histograms shows the count of acoustic emissions.
24Chapter 2. Coda Wave Interferometry for Estimating Nonlinear Behavior in Seismic Velocity
is measured for a temperature change of about 100 ◦ C (Hughes & Maurette, 1956; Peselnick
& Stewart, 1975; Kern et al., 2001). In our experiment a cylindrical sample of granite with a
height of 110 mm and a diameter of 55 mm was heated with a heating coil inside the sample
from 20◦ C to 90◦ C and cooled down to room temperature. The heating and cooling phase
took about 8 hours each. Two piezo-electric transducers were used to excite and record
elastic waves in the sample with a dominant frequency of about 100 kHz. The waveforms
were recorded after each ±5◦ C change in temperature. In order to reduce the influence of
ambient noise the wave-forms were stacked over 10 shots. A third transducer was used to
monitor the acoustic emissions in the sample.
The difference in the early part of the wave-forms recorded at temperatures of 45 ◦ C
and 50◦ C (Figure 2.4) are small. This change in temperature does not affect the first arrival,
which means that the travel time of the first arrival cannot be used to infer any possible
small change in velocity due to a 5◦ C temperature difference. The late time window (bottom
inset) shows a clear time shift of the wave-forms.
For each change of ±5◦ C in temperature the change in the velocity is inferred from
equation (2.10) using 20 different time windows of the coda waves with a duration of 0.1
ms each. The mean and variance of the velocity change (Figure 2.4) is inferred from the
estimates of the the velocity change in the different time windows. The relative velocity
change is of the order of 0.1% for a temperature change of ±5 ◦ C with an error of about
0.02%.
During the heating phase the velocity change is constant for temperatures less than
75◦ C. Above that temperature the velocity change during heating increases (Figure 2.4).
The acoustic emissions correlate with the increased value of the velocity change at 75 ◦ C
(Yong & Wang, 1980). During the cooling phase the velocity change is constant, and there
are no acoustic emissions. When the sample is heated again to a temperature of 90 ◦ C
the velocity change does not increase dramatically around 75 ◦ C and there are no acoustic
emissions (Ide, 1937). In order to test whether the transducer coupling and the presence
of the heating coil played a role we repeated the experiment with an aluminum sample. In
that case the velocity change is constant both during heating and cooling.
The acoustic emissions and the change in the velocity gradient occur only in a pristine
sample during heating (the Kaiser effect (Yong & Wang, 1980) and are due to the irreversible formation of fractures by differential thermal expansion (Meredith et al., 2001) of
the minerals in the sample. This indicates that the velocity change is due to two different
mechanisms. The first is a reversible change in velocity due to the change in bulk elastic
constants with temperature. The second mechanism is associated with irreversible changes
in the sample that generate acoustic emissions. The damage done to the sample leads to a
greater change in the seismic velocity with increasing temperature.
These measurements could be carried out because of the extreme sensitivity of coda
wave interferometry to changes in the medium. This makes it possible to infer the nonlinear
dependence of the velocity on temperature that is associated with irreversible damage to
the granite sample.
Alexandre A. Grêt
2.5
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Time-Lapse Monitoring with Coda Wave Interferometry
25
Acknowledgments
We thank Robert Kranz and Mike Batzle for their help and advice. This work was partially supported by the NSF(EAR-0106668 and EAR-0111804), by the US Army Research
Office (DAAG55-98-1-0070), and by the sponsors of the Consortium Project on Seismic
Inverse Methods for Complex Structures at the Center for Wave Phenomena.
26Chapter 2. Coda Wave Interferometry for Estimating Nonlinear Behavior in Seismic Velocity
Alexandre A. Grêt
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Time-Lapse Monitoring with Coda Wave Interferometry
27
Chapter 3
Time-lapse Monitoring of Rock Properties with
Coda Wave Interferometry
3.1
Abstract
The coda of seismic waves consists of that part of the signal after the directly arriving
phases. In a finite medium, or in one that is strongly heterogeneous, the coda is dominated
by waves which have repeatedly sampled the medium. Small changes in a medium which
may have no detectable influence on the first arrivals are amplified by this repeated sampling
and may thus be detectable in the coda. We refer to this use of multiple-sampling coda
waveforms as coda wave interferometry. We have exploited ultrasonic coda waves to monitor
time-varying rock-properties in a laboratory environment. We have studied the dependence
of velocity on uni-axial stress in Berea sandstone, the temperature dependence of velocity in
granite and in aluminum, and the change in velocity due to an increase of water saturation
in sandstone. There are many other possible applications of coda wave interferometry in
geophysics, including dam and volcano monitoring, time-lapse reservoir characterization,
earthquake relocation, stress monitoring in mining and rock physics.
3.2
Introduction
Geophysicists investigate the structure of the subsurface by making indirect measurements on the surface and relating them to those predicted by theoretical Earth models.
The Earth, however, is a highly complex system, and we almost always have to simplify our
models in order to make them tractable. In many applications, this simplification means
treating unmodeled physics as noise, with the result that information contained in the data
is discarded. For seismic data, this typically means ignoring the coda waves that make up
the tail of a seismogram. (In music the coda is the concluding passage of a movement or
composition (Latin cauda, tail).) Geophysical applications based on use of the coda waves
include proposed schemes for earthquake prediction (Aki, 1985; Sato, 1988), volcano monitoring (Poupinet et al., 1984; Ratdomopurbo & Poupinet, 1995; Fehler et al., 1998; Aki
& Ferrazzini, 2000) or monitoring of temporal changes in the subsurface (Chouet, 1979;
Revenaugh, 1995; Baisch & Bokelmann, 2001; Niu et al., 2003).
Consider the following examples: in monitoring a nuclear waste disposal site, one is not
primarily interested in imaging the site. It is, however, critical to monitor temporal changes
in the site. In recent years, applied geophysicists have spent much effort on time-lapse
28 Chapter 3. Time-lapse Monitoring of Rock Properties with Coda Wave Interferometry
seismology to monitor hydrocarbon reservoirs during recovery operations. Hydrocarbons
move in the subsurface, reservoir rocks are artificially fractured, water-oil horizons move
and injected steam propagates through the reservoir (Lumley, 1995; Wang, 1997). The high
sensitivity of coda waves to small perturbations of the medium makes them a powerful tool
to monitor these kinds of changes.
We present four laboratory experiments in which we monitor the change in seismic
velocity resulting from (i) a change in uni-axial stress in a sample of Berea sandstone, (ii)
a change in water saturation in a sample of Berea sandstone, (iii) a temperature change
in a sample of aluminum and (iv) in a sample of Elberton granite. We excited and record
ultrasonic waves to extract the velocity change from the coda waves.
3.3
Sensitivity of coda waves
In a tomographic transmission experiment, the area under investigation is usually
sampled once. The traversing waves have a certain sensitivity to a velocity change in
this area (depending on distance, velocity and sampling). In a coincident-source-receiver
reflection experiment the area traversed by the waves is sampled twice and is therefore twice
as sensitive to a velocity change than in the transmission case. Hence, in a setup where a
wave is bouncing back and forth, and samples the same area multiple times, the wave is
much more sensitive to a velocity change. This amplification of a velocity change, due to
the multiple sampling of the same area is the key idea we use in this research.
Figure 3.1 outlines some examples of wave propagation where multiple sampling of the
same area is achieved. In figure 3.1a the waves bounce back and forth between opposite
boundaries of the medium, we think of this as a “bouncing ball mode.” In the laboratory
experiment where we monitor a velocity change due to a change in uni-axial stress, we record
a wave field which is dominated by such a bouncing ball mode (see “Monitoring uni-axial
stress in Berea sandstone”). In figure 3.1b surface waves sample the medium by circling
around the boundary, they are influenced by the velocity change each time they encircle
the medium. Because of the cylindrical symmetry of the samples, surface waves dominate
the coda in the experiment where we monitor temperature in aluminum and granite (see
“Monitoring thermally induced velocity changes in aluminum” and “Monitoring thermally
induced change and acoustic emissions in granite”). In figure 3.1c waves propagate in a more
complex manner, bouncing off the boundaries in a complex pattern. Because of the irregular
shape of the sandstone sample in the experiment where we monitor water saturation, we
record such complex waveforms (see “Monitoring water saturation in Berea sandstone”).
If the medium under investigation contains many small-scale scatterers, and the wave
follows a path that connects these scatterers (figure 3.1d), then the scattered path is much
longer than the direct path from source to receiver. Hence the scattered wave is more sensitive to a velocity change than the direct wave. Examples of such scatterers in a background
medium are colloidal suspensions, aerosols in air, and grains in rock.
In a laboratory environment it is difficult to avoid waves that repeatedly sample the
medium, because the samples are of finite size. The associated repeated sampling (as in
Figure 3.1a) lends itself naturally to a modal (i.e. resonance) analysis, as shown for rock
Alexandre A. Grêt
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Time-Lapse Monitoring with Coda Wave Interferometry
a
b
c
d
29
Figure 3.1. Cartoon of different wave modes that may co-exist in a medium. a) A bouncing
ball mode where the waves go straight back and forth between two boundaries. b) Surface
waves that propagate along the boundary, circling the medium. c) More complex reverberations between the boundaries. d) Multiple scattering from small-scale scatterers (small
circles) in the medium.
30 Chapter 3. Time-lapse Monitoring of Rock Properties with Coda Wave Interferometry
cores by Scales & Malcolm (2003). Modal analysis is used in long period seismology (e.g.,
Dahlen & Tromp, 1998)) and resonance spectroscopy (McSkimmin, 1964; Zadler et al.,
2004). The power of resonance comes from the ability to use frequency to selectively excite
certain waves (e.g., a bouncing ball modes), and from Rayleigh’s Principle (Rayleigh, 1896,
chap. 88) which states that the perturbation in the squared eigenfrequency of a mode can
be related to perturbations in the density and elastic moduli, involving only the unperturbed
mode vectors.
In field situations, however, there will likely be many scenarios where no modes can
be excited (Figure 3.1d). In those situations, coda wave interferometry can still be applied
as well as in the case where modes are excited.
This sensitivity increase with longer path length is extensively used in optical applications, for example in atmospheric pollution analysis. In this application, air is introduced
into a long tube. A laser pulse propagates trough the tube parallel to the long axis of the
tube, and is measured at the other end. The ratio of the incident laser energy to the outgoing energy is a function of tube length, pollution density and pollution type. The longer
the tube (path length), the more sensitive the instrument (Hodges et al., 2004). Similar to
coda wave interferometry, the path can be increased by multiple sampling of the same area.
In optics this is called Cavity Ring-Down Spectroscopy (O’Keefe & Deacon, 1988).
3.4
Estimation of velocity change from coda waves
For a constant change δv in seismic velocity and fixed location of the scatterers and
reflectors, we can write the propagation path l = vt, where v is the constant seismic velocity
and t the propagation time. For a homogeneous velocity change δv in the medium and an
unchanged path we get l = vt = (v + δv)(t + δt), or to first order
−δt
δv
=
,
(3.1)
v
t
where δt is the travel-time difference caused by the velocity change δv. A more rigorous derivation in given in the appendix. We extract δt from the data by means of the
cross-correlation function, where δt is given by the position of the maximum of the crosscorrelation function that is defined as
R t+tw
uunp (t0 ) uper (t0 + ts )dt0
R(t,tw ) (ts ) ≡ R t+tt−tw
R t+t
1 ,
( t−tww u2unp (t0 )dt0 t−tww u2per (t0 )dt0 ) 2
(3.2)
where the time window is centered at time t with duration 2t w , ts is the time shift used in
the cross-correlation, uunp is the unperturbed (before the velocity change) wave field and
uper the perturbed wave field.
In the following experiments where we monitor rock properties, the coda consists of
waves that are repeatetly reflected from the free surface, and waves scattered from the
crystals and grains in rocks (see previous paragraph). The above formulation (equation 4.1,
using the path summation in the appendix), includes all the scenarios for multiple scattering
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Time-Lapse Monitoring with Coda Wave Interferometry
31
sketched in figure 3.1, and we can use the same theory for all our measurements.
It is important to note that up to this point we have not distinguished between P- or
S-waves. An extension for elastic waves is given by Snieder (2002), who uses an analysis
based on a simple P- and S-wave equilibration model. Snieder (2002) shows that coda wave
interferometry is much more sensitive to S-wave than P-wave velocity, in particular for a
Poisson medium the velocity change measured with coda wave interferometry is given by
δv
δvP
δvS
≈ 0.09
+ 0.91
,
v
vP
vS
(3.3)
where vS and vP are S- and P-wave velocities, and δv S and δvP are the velocity changes
for S and P-waves, respectively. In some situations, the coda can be dominated by a modal
behavior as sketched in figure 3.1. In the case of a surface wave that repeatedly propagate
around the circumference of a sample (figure 3.1b), we measure a change in surface wave
velocity, which is approximately 0.9 times the S-wave velocity for typical Rocks (Carmichael,
1982).
In the following experiments we only consider the relative velocity change estimated
from the coda of the ultrasonic measurements. Other types of perturbations such as the
displacement of the source or receiver position and opening and closing of pores and fractures, leave a different signature on the time shifted correlation coefficient than a constant
velocity change. Snieder (2004) gives a summary and a brief explanation for the different
types of change that could potentially be monitored with coda wave interferometry.
All these laboratory experiments involve essentially the same measurement of ultrasonic waves, we measure the impulse response of a rock sample with compressional ultrasonic
transducers. The difference between the experiments consists of the physics of the change
introduced (stress, fluid saturation or temperature), the geometry, size and type (Berea
sandstone, Elberton granite and aluminum) of the samples.
3.5
Monitoring uni-axial stress in Berea sandstone
Time-varying stress fields are important in a number of areas of geophysics. Changes
near plate boundaries are important in order to understand plate tectonics (Bokelmann &
Silver, 2002). In earthquake prediction, the stress field is important for understanding fault
behavior and its relation to earthquake occurrence (Stein, 1999; Freed & Lin, 2001; Niu et al.,
2003). In hydrocarbon reservoirs, the stress field is changed by recovery operations. It is
important to understand the associated temporal change for time-lapse reservoir monitoring
(Teanby et al., 2004). In underground coal mining, “Room and Pillar” is a method in which
approximately half of the coal is left in place to support the roof of the active mining area.
Monitoring the stress field in the pillars and roofs is crucial for safe mining operations
(Nikitin, 2003).
Wyllie et al. (1958) measured ultrasonic P -wave velocity as a function of effective stress
in water-saturated Berea sandstone. They showed that at a constant confining pressure, v p
increases with decreasing pore pressure, and for constant effective stress the velocity remains
approximately constant. Similar relationships between effective stress and P -wave velocity
32 Chapter 3. Time-lapse Monitoring of Rock Properties with Coda Wave Interferometry
s
y
y
x
z
Figure 3.2. The first experiment a 12.5 x 40 x 40 cm 3 block of Berea sandstone is
subjected to a uni-axial load in the y-direction. Ultrasonic waves propagate in the
z-direction.
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Time-Lapse Monitoring with Coda Wave Interferometry
33
−3
x 10
0.02
a)
4
0.01
b)
2
0
0
−0.01
−2
−0.02
0.5
1
time [msec]
1.5
−4
0.6
0.65
0.7
time [msec]
0.75
Figure 3.3. a) Two waveforms recorded at an applied uni-axial load level of 6 MPa (thick
line) and 8 MPa (thin line). b) The same two waveforms as in figure 3.3a, but only a small
time window of the signal is shown; the time interval is marked by the two dashed lines in
3.3a. The path length of the ultrasonic wave in this time interval is about 2.5 m and the
wave has bounced back and forth about 20 times.
have also been reported by others (Nur & Simmons, 1969; Hicks & Berry, 1956; King, 1966;
Christensen & Wang, 1985, e.g.,). Hence, a good knowledge of seismic velocity is important
for pore pressure prediction. Experimental results indicate that confining and pore pressures
have almost equal but opposite effects on v p (Terzaghi’s effective stress principle). Confining
pressure influences the wave velocities because pressure deforms most of the compliant parts
of the pore space, such as micro-cracks and loose grain contacts. Closure of micro-cracks
increases the stiffness of the rock and increases bulk and shear moduli. An increase in pore
pressure mechanically opposes the closing of cracks and grain contacts, which lowers the
effective moduli and velocities. Hence, when both confining pressure and pore pressures
vary, only the difference between the two pressures has a significant influence on velocity
(Terzaghi, 1936; Hicks & Berry, 1956).
We show how coda wave interferometry can be used for stress-field monitoring in
a laboratory environment. In this experiment we use a fine-grained Berea sandstone to
investigate the dependence of elastic waves on uni-axial stress. The sandstone block is
equipped with an ultrasonic source (transducer) on one side and a receiver on the other
(Figure 3.2). The transducer excites the rock with a pulse with a dominant frequency of
0.2 MHz. A single receiver records the propagated waves, with a sampling interval of 1 µs.
To reduce the noise level, 512 traces are stacked for each stress level. A typical record is
shown in Figure 3.3a.
To introduce a controlled change in the medium over time, the sandstone block is
placed in a hydraulic press and a uni-axial load is applied (see Figure 3.2). We monitor
34 Chapter 3. Time-lapse Monitoring of Rock Properties with Coda Wave Interferometry
0.03
0.025
δv/v
0.02
0.015
0.01
0.005
0
0.2
0.4
0.6
0.8
1
window center time [msec]
1.2
Figure 3.4. Velocity change estimates for 20 non-overlapping windows with different center
times. The mean velocity change is 1.02 % and the standard deviation is 0.16 %.
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Time-Lapse Monitoring with Coda Wave Interferometry
35
uni-axial load by a pressure sensor between rock and press. For each stress state (4, 6 and
8 MPa) the ultra-sonic measurement is repeated.
Figure 3.3a shows two waveforms superimposed, one at a load of 6 MPa and the
other at 8 MPa. After about 0.5 ms, the waves have a noisy appearance. If we look at a
smaller time-window as shown in figure 3.3b, we see a strong correlation between the two
waveforms, with one waveform time-shifted with respect to the other. Thus, despite the
noisy appearance of the coda waves, these waves carry information about the structure of
the medium, information that can be used to infer the change of sonic velocity with applied
pressure.
We infer the velocity change caused by an increase in the load from 6 MPa to 8MPa,
from the phase shift in the coda waves, using 20 non-overlapping time windows each with
a duration of 0.05 ms each of the coda waves, as described in (Snieder et al., 2002). The
windows provide a independent estimates of the relative velocity change. This can be used
for a consistency check of the method. Since we have multiple estimates of δv/v we can
calculate the mean and variance of the relative velocity change. The relative velocity change
for these time windows is shown in Figure 3.4, this change is of the order of 1.02% for a
increase in load of 2 MPa with an error of 0.16%.
Sarkar et al. (2003) shows comparable velocity changes in the same Berea sandstone
block using one-way travel-times of the first arrivals (1 % for P-waves and 3 % for S-waves).
They estimate the uncertainty, based on errors in travel-time picking to be approximately
1 %. Note that their measured velocity change is on the same order of magnitude than
the estimated uncertainty. The accuracy and sensitivity of coda wave interferometry are
an order of magnitude higher than methods based on one-way travel-time. In contrast to
coda wave interferometry, Sarkar et al. (2003) are able to measure P- and S-wave velocity
independently.
Monitoring stress changes is important, for example, in mining applications, radioactive waste disposal cites or fault zones. Using coda wave interferometry in these applications
could lead to a highly sensitive stress-change monitoring technique with modest hardware
requirements. Grêt et al. (2004b) apply coda wave interferometry to monitor stress changes
in a gold and silver mine.
3.6
Monitoring water saturation in Berea sandstone
Seismic methods can monitor ground water (Bachrach & Nur, 1998), Dense NonAqueous Phase Liquid (DNAPL) contamination movement (Griffin & Watson, 2002), and
hydrocarbon migration (Lumley, 1995; Mjaaland et al., 2001) by detecting changes in seismic velocity. Compressional and shear wave velocities respond to changes in the bulk and
shear modulus and density, caused by the presence of water. We demonstrate the application of coda wave interferometry for monitoring changes in water saturation in Berea
sandstone.
We use an irregularly shaped piece of Berea sandstone with an approximate height of
20 cm and an approximate diameter of 5 cm. The sample is equipped with a compressional
source on one side and a receiver on the other (Figure 3.5). The room-dry sample is placed
36 Chapter 3. Time-lapse Monitoring of Rock Properties with Coda Wave Interferometry
Figure 3.5. The Berea sandstone sample as the fluid is infiltrating. Ultrasonic source and
receiver glued to the rock. The left picture shows the room-dry sample. The middle picture
shows the water rising in the sandstone sample (dark line about a fourth up the sample)
and the right picture shows the water almost half way up the rock. The third transducer
glued to the rock is not used in this experiment.
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Time-Lapse Monitoring with Coda Wave Interferometry
37
0.4
0.2
0.000005
0.00001
0
-0.2
0.000195
-0.4
0
0.0001
0.0002
time(s)
0.0002
0.0003
0.0004
Figure 3.6. Wave-forms recorded in the Berea sandstone block for two different levels of
water saturation (water infiltrated 2 cm of the rock (dashed line) and 3 cm (thin line)).
The insets show details of the wave-forms around the first arrival (top) and in the late coda
(bottom.). The dominant frequency is 0.33 MHz.
38 Chapter 3. Time-lapse Monitoring of Rock Properties with Coda Wave Interferometry
Figure 3.7. Relative velocity change δv/v in Berea sandstone, for approximately 1 cm
increment in water level from room-dry to 9 cm into the sample. Error bars are plus and
minus one standard deviation.
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Time-Lapse Monitoring with Coda Wave Interferometry
39
0.02
0.01
0
-0.01
-0.02
96
99
time (µs)
102
Figure 3.8. Eight waveforms, each measured at a different height of the water front. The
fastest (thickest line) is measured on the dry sandstone and the slowest (thinnest line) is
measured at a water front height of 7 cm. In addition to the velocity change, the amplitudes
decrease with increased water content in the sample.
40 Chapter 3. Time-lapse Monitoring of Rock Properties with Coda Wave Interferometry
in a container that holds 5 mm of water. While the water is infiltrating into the pores of the
sandstone by capillary pressure (Wulff & Mjaaland, 2002), the water-level in the container
is kept constant at 5 mm. When the water-front is rising from 5 mm to 10 cm, we repeat
the ultrasonic impulse-response measurement for every 1 cm increase in water-level in the
sandstone. Again, for a 20 cm sample and a water-front rise of 1 cm there is no significant
travel-time difference for the first arriving waves (see top inset of Figure 3.6). In a late time
window (bottom inset of Figure 3.6), however, we see a distinct time shift of the wave-forms.
Figure 3.8 shows the consistent slowing of the sonic waves with increasing water-level.
We infer the relative change in velocity for each change of 1cm in water-level using
12 different 0.1 ms time windows of the coda waves. The relative velocity change is of
the order of -0.3% for a water-level rise of 1cm with an error of 0.05% (Figure 3.7). It
is important to note that in many laboratory experiments, changes in rock properties are
measured for saturation changes of about 5% on small samples (Spencer, 1981). With coda
wave interferometry we can monitor fluid saturation about 10 times more precisely.
In addition to the effect on velocity, fluid saturation changes the attenuation of the
rock (Mavko & Nur, 1979). In figure 3.8 one can clearly see the amplitude decrease with
increased water-level. When monitoring velocity changes with coda wave interferometry,
this attenuation change doesn’t influence the estimates of the velocity change, because the
normalization used in equation (3.2) renders the quantity independent of amplitude changes.
3.7
Monitoring thermally induced velocity changes in aluminum
The dependence of ultrasonic velocity on temperature in metals and alloys is an important characteristic in non-destructive testing (Kobori & Iwashimizu, 1990, e.g.,). Often,
the effect of stress on this velocity/temperature relationship is studied (Salama & Ling,
1980; Chern & Heyman, 1981). Multiply scattered or reverberating waves are known to be
sensitive to variations in temperature (Weaver & Lobkis, 2000). We use this sensitivity of
coda waves to monitor temperature changes in aluminum.
In this ultrasonic experiment we use an aluminum cylinder with a height of 11 cm and
a diameter of 5.5 cm. The sample is equipped with an ultrasonic source on one side and
a receiver on the other (Figure 3.9). The transducer sends a pulse through the sample,
and the single receiver records the impulse response of the sample, with a sampling interval
of 1µs (the dominant frequency is 100 kHz.) We stack ten traces to reduce the noise
level. Two typical records for a cylindrical sample are shown in Figure 3.10. To apply a
controlled change in the medium, the aluminum sample is equipped with a heating element
in a central borehole. We monitor the temperature with two thermocouples glued to the
side of the sample and in the borehole (Figure 3.9).
While increasing the temperature from 25 o C to 90o C, we repeat the ultrasonic measurement for every 5o C increase in temperature. Then the aluminum sample is cooled
to room-temperature and we repeat the ultrasound measurement again for every 5 o C in
temperature decrease.
In some published laboratory experiments, the change in the seismic velocity is measured for a temperature change of about 100 o C (Timur, 1977; Peselnick & Stewart, 1975;
Alexandre A. Grêt
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Time-Lapse Monitoring with Coda Wave Interferometry
41
Heating Coil
Ultrasonic Receiver
Thermo Couple (Temperature)
Acoustic Emissions
Ultrasonic Source
Coda Waves
Ultrasonic Receiver
Figure 3.9. The cylinder represents the Elberton granite or the aluminum sample. Sonic
waves are transmitted through the sample. A longitudinal transducer, which excites primarily P-waves, and an identical receiver (right rectangle) are used throughout the experiment.
A third identical P-wave transducer (top left rectangle) detects acoustic emissions. The
sample is heated with a heating coil placed in a centered borehole and the temperature is
measured with a thermo-couple at the sample surface (white rectangle.)
42 Chapter 3. Time-lapse Monitoring of Rock Properties with Coda Wave Interferometry
2
1
0
1
-1
0
0.005
0.01
0.015
0.46
0.465
0.47
0.475
0
0.4
0.2
-1
0
-0.2
0
0.5
1
time (ms)
1.5
2
Figure 3.10. Wave-forms recorded in the granite sample for temperatures of 45 o C (thin
line) and 50o C (thick line), respectively. The insets show details of the wave-forms around
the first arrival (top) and in the late coda (bottom.) The main frequency in the data is 0.13
MHz.
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Time-Lapse Monitoring with Coda Wave Interferometry
43
1.5
0.5
−0.5
−1.5
|dv/v|
0.003
0.002
0.001
0.000
0.0
0.5
1.0
Time (ms)
1.5
2.0
Figure 3.11. The top figure shows the ultra-sonic signal recorded on the aluminum sample.
The bottom figure shows different estimates of δv/v for multiple time-windows, therefore
providing a consistency check.
44 Chapter 3. Time-lapse Monitoring of Rock Properties with Coda Wave Interferometry
0.003
|dv/v|
0.002
0.001
0.000
20
40
60
80
Temperature (Celsius)
100
Figure 3.12. Absolute values of δv/v in aluminum, for 5 o C temperature intervals from
25o C to 90o C. Circles correspond to the heating phase (velocity decrease) and rectangles
(unfilled) to the cooling phase (velocity increase).
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Time-Lapse Monitoring with Coda Wave Interferometry
45
Hughes & Maurette, 1956). For a 11 cm small sample and a temperature difference of only
5o C, there is no significant travel-time difference for the first arriving waves (see top inset of
Figure 3.10). Therefore, first arriving waves do not provide any information about velocity
changes for such a small temperature difference. In a late time window (bottom inset of
Figure 3.10), we see a distinct time shift of the wave-forms. This information can be used
to infer the change of ultrasonic velocity with temperature.
We estimate the relative change in velocity for each change of 5 o C in temperature
with 20 different 0.1 ms time windows of the coda waves (figure 3.11). The relative velocity
change is of the order of 0.15% for a temperature change of 5 o C with an error of 0.025%
(Figure 3.12). It is important to note that with the exception of a sign change, the relative
velocity change with temperature does not depend on whether the sample is in the heating
or the cooling phase. In other words, if we sum all the relative velocity changes for the
heating phase (negative velocity change) and the cooling phase (positive velocity change)
we obtain a relative velocity change after the heating cycle that is approximately zero.
We use this laboratory experiment to test the presence of non-linear temperature
effects on the measurement equipment, such as the piezoelectric transducers, the cables, the
transducer couplant, and mounting devices. We measure a linear dependence of velocity on
temperature in aluminum as in the experiment of Weaver & Lobkis (2000). We therefore
conclude that non-linear instrument effects can be neglected. The change in path length
due to thermal expansion of aluminum (10 −5 C −1 (Carmichael, 1982)) causes an apparent
velocity change. This effect is two orders of magnitude smaller than the obtained velocity
change from the coda waves and can be neglected.
3.8
Monitoring thermally induced velocity change and acoustic emissions in
granite
With the same technique and same experimental setup as described in the previous
section, we measured the thermally induced velocity change in a granite sample. In addition,
we count acoustic emissions for every temperature interval. Since the coefficient of thermal
expansion for quartz is 10−6 C −1 (Carmichael, 1982), the associated apparent velocity change
is three orders of magnitude smaller than the velocity change obtained from the coda waves
and can be neglected.
During the heating phase the velocity decrease (negative relative velocity change) is
constant for each 5o C increase in temperatures, for temperatures below 70 o C. At that
temperature, however, the velocity change is non-linear (Figure 3.13). Since we tested for
non-linear temperatrure effects of the measurement equipment on the aluminum sample,
the non-linear velocity decrease in granite must be attributed to a change in the rock sample. The temperature of 70o C corresponds to the critical fracture temperature for granite
(Johnson et al., 1978; Fredrich & Wong, 1986). Thermal cracking results from the internal
stress concentration induced by thermal expansion anisotropy or thermal expansion mismatch between minerals or grains. Such micro-cracking is a similar effect as the thermal
stresses induced by thermal gradients in homogeneous solids; for a high temperature gradient, cracking may occur even in a perfectly homogeneous solid (Boley & Weiner, 1960).
0.004
4000
0.003
3000
0.002
2000
0.001
1000
0.000
20
40
60
80
Temperature (degree Celsius)
Counts
| dv/v |
46 Chapter 3. Time-lapse Monitoring of Rock Properties with Coda Wave Interferometry
0
100
Figure 3.13. Absolute values of δv/v in Elberton granite, for 5 o C temperature intervals
from 25o C to 90o C. Circles correspond to the heating phase and rectangles to the cooling
phase. The histograms show the count of acoustic emissions for a given temperature interval
during the heating phase.
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Time-Lapse Monitoring with Coda Wave Interferometry
47
Velocity (m/s)
5000
4950
4900
4850
20
40
60
80
Temperature (Celsius)
100
Figure 3.14. Velocity versus temperature in Elberton granite, for two heating cycles. Filled
circles represent the first heating cycle and open rectangles the second. Note that during the
second heating cycle the temperature dependent velocities during the heating and cooling
phase are indistinguishable.
48 Chapter 3. Time-lapse Monitoring of Rock Properties with Coda Wave Interferometry
Fredrich & Wong (1986) show that thermal cracking in rocks occurs principally along mineral or grain boundaries. The thermally induced cracks can significantly influence both the
mechanical and transport properties, as well as thermoelastic moduli (Simmons & Cooper,
1978).
In this experiment we use a third ultrasonic transducer to detect acoustic emissions
in the granite due to thermal cracking. The histogram in Figure 3.13 shows the count of
acoustic emissions versus temperature. There is a small number of acoustic emissions at
low temperatures. There is, however, a significant increase in acoustic emissions between
70o C and 75o C. The increase in velocity change and the jump in the number of acoustic
emissions correlate well.
Kaiser (1953) found that during repeated loading of metals, little or no acoustic emissions occurred until previously applied stress levels where exceeded. Since then, this effect
has been known as the “Kaiser effect.” Later, it was found that the Kaiser effect is a
common phenomenon for various materials including rocks (Kurita & Fujii, 1979; Lavrov,
2002). Thus, the maximum stress applied in the previous cycles is ‘memorized’ in rocks.
During the cooling phase of the granite, the velocity depends linearly on temperature
over the whole temperature change and there are few acoustic emissions. The seismic
velocity does not return to its initial value at the end of the cycle. This difference in
velocity is due to irreversible damage done to the rock by thermal cracking (Figure 3.14).
Todd (1973) studied the acoustic emissions of Westerly granite during cyclic heating.
He noted that if a sample was re-heated to the same maximum temperature, few acoustic
emissions occurred. Similarly we find in a second heating cycle up to the same maximum
temperature (90o C) for the same granite sample, only few acoustic emissions occur and
there is no non-linear velocity decrease around 70 o C. Furthermore, the velocity increases
back to the value before the second heating cycle when cooled down (Figure 3.14). Note
that there is a small difference in relative velocity change between the cooling phase of the
first heating cycle and the second cycle. Thirumalai & Demou (1973) studied the residual
strain in a granitic rock produced by cyclic heating, and showed that predominant damage
took place during the initial exposure to heating and the damage reached a steady state
after three successive heating cycles. If we increase the temperature above the previous
maximum temperature (90o C), the same non-linear effect occurs; the granite “remembers”
the maximum temperature.
This indicates that two different mechanisms drive the temperature induced velocity
change. The first mechanism is the change in bulk elastic constants with temperature,
which is linear and reversible. This explains the constant velocity change with temperature
during the second heating cycle during heating and cooling. The second mechanism is the
irreversible damage done to the granite due to thermal cracking, which explains the nonlinear velocity change at the critical fracture temperature during the first heating cycle.
Ide (1937), found the same temperature dependence of velocity on temperature in
Quincy granite. Using the travel time of first arrivals, he obtained 7 measurements over
one heating cycle, with a peak temperature of 300 o C. With coda wave interferometry we
are able to measure twenty times more points over the same temperature interval. In other
words, coda wave interferometry is orders of magnitude more sensitive to a temperature
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Time-Lapse Monitoring with Coda Wave Interferometry
49
change.
3.9
Conclusions
Due to the sensitivity of coda waves, we are able to study the influence of changes in
stress, temperature and fluid on small samples to a high level of precision. The key idea
is that multiple sampling of the same area increases the sensitivity of coda waves. This
new level of sensitivity may lead to a better understanding of rock properties and material
properties in general.
The velocity estimation based on the coda waves requires only a single repeatable
source and a single receiver, which makes it a potential method for inexpensive, real-time
monitoring of land slides, hydrocarbon reservoirs, volcanoes, nuclear waste disposal sites
and as a diagnostic tool in non-destructive testing. Snieder & Vrijlandt (2004) apply coda
wave interferometry to obtain the relative location of earthquakes from the seismic coda.
Coda wave interferometry has similarities with Cavity Ring-Down spectroscopy (O’Keefe
& Deacon, 1988) and resonance spectroscopy (McSkimmin, 1964). In those methods modes
are excited and analyzed in time or frequency. In contrast, we can use coda wave interferometry in field experiments, when it is not possible to excite modes because the system is
open. For example, we use multiply scattered seismic waves to monitor a rapid temporal
change in a volcano (Grêt et al., 2004a). We could use coda wave interferometry to monitor
minute changes in-situ, for example in groundwater monitoring or DNAPL contamination.
On a global seismology scale, coda wave interferometry could play an important part in
monitoring large scale changes using surface energy from earthquakes that circle the earth
many times (Earth HUM). In addition there are countless industrial applications, where
monitoring of stress, cracks, temperature and fluids is important, including aircraft engine monitoring, monitoring of punch press heads or applications in medical imaging like
monitoring osteoporosis.
3.10
Acknowledgments
We thank Bob Kranz, Debashish Sarkar and Mike Batzle for their help with the experiments. We also thank the members of the Center for Wave Phenomena and the Physical
Acoustics Laboratory for stimulating discussions. This work was partially supported by the
NSF(EAR-0106668 and EAR-0337379).
3.11
Appendix
This appendix follows Snieder (2002). For a change in the wave velocity, for quasirandom perturbations of the point scatterer location, or for a change in the source location,
we can estimate this perturbation from multiply scattered waves by a cross-correlation in
the time domain (Snieder et al., 2002). We refer to the waveform before the perturbation
as the unperturbed signal, and to the waveform after the perturbation as the perturbed
50 Chapter 3. Time-lapse Monitoring of Rock Properties with Coda Wave Interferometry
signal. The unperturbed wave-field can be written as a Feynman path summation over all
possible trajectories T (Snieder, 1999):
uunp (t) =
X
AT (t),
(3.4)
T
where a trajectory is defined as a sequence of scatterers encountered by the wave. The sum
over trajectories contains a sum over all possible mode conversions (P-waves, S-waves and
surface waves), and it describes both trajectories that bounce off the free surface and/or
trajectories that connect scatterers.
When the background velocity is perturbed, the dominant effect on the waveform arises
from the change in the travel time τT of the wave that travels along each trajectory:
uper (t) =
X
T
AT (t − τT ).
(3.5)
We can compute the time-windowed correlation coefficient between the unperturbed and
the perturbed signal from equation (3.2). When the perturbed and unperturbed wave fields
defined by equations (3.4) and (3.5) are inserted into (3.2), double sums over all trajectories
appear. The cross-terms with different trajectories (T 6= T 0 ) are incoherent and average
out to zero when the mean of the source signal vanishes. We therefore approximate the
time-windowed correlation coefficient by:
R(t,tw ) (ts ) ≈
P
T (t,tw )
P
CT (τT − ts )
T (t,tw )
CT (0)
,
(3.6)
where the sum is taken over the trajectories with arrival times within the time window of
the cross-correlation, and the auto-correlation of the source signal is defined as
CT (t) ≡
Z
∞
AT (t0 + t)AT (t0 )dt0 .
(3.7)
−∞
For time shifts τ much smaller than the dominant period, a second-order Taylor expansion gives C(τ ) = C(0)(1 − 12 ω 2 τ 2 ), where ω 2 is the mean-squared frequency of the
waves that arrive in the time window defined as:
2
ω ≡−
P R t+tw
0
0
0
t−tw AT (t )ÄT (t )dt
.
P R t+tw
0 2 0
T t−tw AT (t ) dt
T
(3.8)
Using this in equation (3.6) we can write
where h· · · i(t,tw )
(t − tw , t + tw ).
1 (3.9)
R(t,tw ) (ts ) = 1 − ω 2 (τ − ts )2 (t,tw ) ,
2
stands for the average over the wave paths with arrivals in the time interval
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Time-Lapse Monitoring with Coda Wave Interferometry
51
The time shifted cross-correlation R (t,tw ) (ts ) has a maximum when
ts = tmax ≡ hτ i(t,tw ) ,
(3.10)
where hτ i(t,tw ) is the mean travel time perturbation of the arrivals in the time window.
Using expressions (3.9) and (3.10) gives the maximum value of the cross-correlation
1
(t,tw )
Rmax
= 1 − ω 2 στ2 ,
2
(3.11)
where στ2 is the variance of the travel time perturbation for waves arriving within the time
window. This means that we can extract the mean and the variance of the travel time
perturbations of the waves arriving in a time window.
For a constant change δv in seismic velocity and fixed locations of the scatterers, the
mean travel time perturbation is given by hτ i (t,tw ) = −(δv/v)t. When the time window is
small (tw t), στ ≈ 0. The velocity change follows from the time of the maximum of the
time-shifted cross-correlation function:
−tmax
δv
=
.
v
t
(3.12)
52 Chapter 3. Time-lapse Monitoring of Rock Properties with Coda Wave Interferometry
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Time-Lapse Monitoring with Coda Wave Interferometry
53
Chapter 4
Monitoring stress change in an underground
mining environment with coda wave
interferometry
4.1
Abstract
We developed a new technique (coda wave interferometry) that uses multiply scattered
waves to monitor changes in a medium. Coda waves are highly sensitive to changes in stress,
deformation, temperature, fluids and more. Because of this sensitivity we can monitor small
stress changes in an underground mine. We applied coda wave interferometry to seismic
data excited by a hammer source, collected at an experimental hard rock mine, Idaho
Springs, CO. We carried out a controlled stress-change experiment in a pillar and we were
able to monitor the internal stress change with coda wave interferometry.
4.2
Introduction
The coda is the tail (Latin cauda) of a waveform, it consists of that part of the signal after the directly arriving phases (Aki, 1969). Its late part is dominated by multiply
scattered and/or multiply reflected waves (Aki & Chouet, 1975). Coda waves are used in a
multitude of geophysical applications, such as earthquake-magnitude estimation (Lee et al.,
1972), earthquake prediction (Aki, 1985; Sato, 1986), volcano monitoring (Aki & Ferrazzini, 2000; Fehler et al., 1998), monitoring of temporal changes in the subsurface (Chouet,
1979; Poupinet et al., 1984; Robinson, 1987), and relocation of earthquake clusters (Snieder
& Vrijlandt, 2004). Laboratory applications include diffusive wave spectroscopy (Cowan
et al., 2002), reversed time imaging (Fink, 1997), medical imaging (Li et al., 1997) and
cavity ring-down spectroscopy (O’Keefe & Deacon, 1988).
Changes in a medium can be so small that they have no detectable influence on seismic
waves that sample the area only once. These changes are, however, amplified by multiple
scattering and may be readily seen in the coda. We have developed a new method called
coda wave interferometry (Snieder et al., 2002) to extract information about the change in
the medium from the coda. We have previously used coda wave interferometry to study
the non-linear temperature dependence of ultrasonic velocity in granite (Snieder et al.,
2002), the dependence of ultrasonic velocity on stress change and water saturation in Berea
sandstone (Grêt et al., 2004c), and monitoring of a rapid temporal change in a volcano (Grêt
et al., 2004a). Of particular interest for the mining community is our experiment where
54Chapter 4. Monitoring stress change in an underground mining environment with coda wave interferometry
we monitor acoustic emissions (due to cracking) in granite with coda wave interferometry
(Grêt et al., 2004c).
In this work we demonstrate the usefulness of coda wave interferometry for monitoring
stress changes in an underground mining environment. It is known that the sensitivity
of seismic wave velocity to stress changes in rocks is low (Nur, 1971) and detection of
temporal variations has been extremely difficult (Niu et al., 2003). Attempts have been
made to monitor stress changes in underground mines with travel-time tomography, where
tomographic images based on one-way travel-times of seismic waves for different stress states
are compared (Friedel et al., 1995, 1996). Since a major stress change in rocks only leads to a
small change in velocity, those methods are inherently prone to large uncertainties. Hence,
tomography studies to monitor stress changes in rocks have to be viewed with caution.
Instead of sampling the regional stress once as in the tomography approach, we make use
of the fact that repetitive sampling of the same area by coda waves makes these waves
increasingly sensitive to stress changes. Since coda waves are so sensitive to small changes
in the medium, they are a suitable tool to monitor stress changes in rocks (Grêt et al.,
2004c).
4.3
Short introduction to coda wave interferometry
A detailed description of coda wave interferometry is given by Snieder (2002), and
by Grêt et al. (2004c). Here we give the main results that are needed to understand the
technique. For a change in the medium that consists of a constant change δv in seismic
velocity and unchanged locations of the scatterers and reflectors, we can write the path
length l that the seismic waves cover as l = vt, where v is the seismic velocity and t the
propagation time. For a constant velocity change δv and an unchanged path length l we
get l = vt = (v + δv)(t + δt), hence to first order
−δt
δv
=
,
(4.1)
v
t
where δt is the travel-time difference (in a given time window) caused by the velocity change
δv. We extract δt from the data by means of the cross-correlation function, that is defined
as
R
(t,tw )
R t+tw
uunp (t0 ) uper (t0 + ts )dt0
(ts ) ≡ R t+tt−tw
R t+t
1 ,
( t−tww u2unp (t0 )dt0 t−tww u2per (t0 )dt0 ) 2
(4.2)
where the time window is centered at time t with duration 2t w , ts is the time shift used in
the cross-correlation, uunp is the unperturbed (before the velocity change) wave field and
uper the perturbed wave field (after the velocity change), δt is the arrival time difference
between the waves recorded before the velocity change and the waves recorded after the
velocity change, for all the waves that arrive in the same time window. Hence, we can
estimate δt for multiple non-overlapping time-windows independently and from equation
(4.1) we calculate the relative velocity change for each time window. Thus, coda wave
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Time-Lapse Monitoring with Coda Wave Interferometry
55
interferometry features its own consistency check.
4.4
Experimental setup and local geology
The Edgar Mine, Idaho Springs, CO, is located about 55 km west of Denver, CO, and
is owned and operated by the Colorado School of Mines (CSM), Golden, CO. The 305 m of
crosscuts and drifts that access several silver-gold veins are located at en elevation of 2,405
m. Widths of the crosscuts and drifts average about 3 m, and the overburden above the
mine is about 120 m. The experiment is performed on a 43 m long and 23 m wide pillar
(figure 4.1). In order to introduce a controlled stress change in the pillar, a pressure cell
was installed into the rock by Scott et al. (1999). The slot cut for the pressure cell is 1 m
high and 0.76 m wide and is cut 3 m deep into the pillar. The slot was formed by drilling
closely spaced holes into the pillar, and the pressure cell is installed to a depth of 2.1 m.
Grout was pumped around the pressure cell to allow it to press against the walls of the slot
without expanding excessively. The pressure cell that is used to induce pressure against the
walls of the slot, measures 30 x 30 x 1 cm 3 (inset of Figure 4.1).
The mine is developed in Precambrian metamorphic and granitic rocks. Specifically,
the rocks are assigned to the Idaho Springs Formation. Rocks in the pillar include biotite
schist, biotite microcline pegmatite, biotite-hornblende schist, quartz-feldspar-biotite gneiss
and migmatized gneiss. Ultrasonic core velocity measurements on rock samples from the
Edgar mine were taken in the laboratory by Scott et al. (1999). Ultrasonic P-wave velocities
for the biotite-hornblende-schist range between 5.00 km/s and 5.45 km/s, and for the
pegmatite between 4.72 km/s and 4.75 km/s. Carmichael (1982) lists P-wave velocity
changes for similar rocks (not from the Edgar Mine) for a pressure difference of 1000 psi in
schist and pegmatite that range between 0.03% and 0.045%, depending on the specific kind
and area of origin.
The seismic experiment consists of a repeated hammer source (5 kg sledge hammer)
and two vertical component geophones bolted to the wall on the left and the right of the
slot. One geophone is used as a trigger channel and the other for data acquisition with
a Tektronix digital oscilloscope. The full seismic waveforms are recorded and saved to a
computer.
The pressure in the cell is increased in steps of 220 psi, from 0 to 2200 psi. For
every increase of pressure, we record three repeated seismic waveforms (three hits with
the hammer). Those three seismic records are then stacked (averaged) in order to reduce
random noise in the data. In addition, we apply a bandpass filter with corner frequencies of
20 Hz and 10 kHz to further reduce the electronic noise from the instruments. The stacked
and filtered waveforms, recorded at different pressures in the cell are then compared to
extract information about the stress change in the pillar.
4.5
Reproducible seismic waveforms
For late times, coda waves commonly have a noisy appearance (figure 4.2a). If we look
closer, we find that not only are the first arriving wave modes reproducible (figure 4.2b) but
56Chapter 4. Monitoring stress change in an underground mining environment with coda wave interferometry
R90−1
HT
AIG
TR
S
T
IGH
C−R
Phone
HI GH E
TAG
VOL
R90
UR
SP
nd
B−R2
RI
B−
T
GH
Slot with pressure cell
(3.04 m into rib,
1.83 m high,
7.62 cm wide)
R30−3
T
GH
RI
−
B
T
GH
AI
R
ST
R30−2A
Slot with Flat Jacks
Hammer source
TP−1
HI
VO
Geophone
GH GE
L TA
T
GH
RI
B−
BRRTEMP
R−170
0
R−180
Scale, m
20
10
30
R−190
Figure 4.1. Plan view of the Edgar Mine pillar. Geophone and hammer source locations
are labeled in the plan. The pressure cell in the slot is indicated by the small line between
geophone and source location. The inset (top right) sketches the pressure cell installed in
the pillar.
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Time-Lapse Monitoring with Coda Wave Interferometry
57
0.01
0.003
0
0.005
-0.003
-0.001
0
0.001
0.002
0.003
0.102
0.103
0.104
0.105
0
0.0003
-0.005
0
-0.0003
0.101
0
0.05
time (s)
0.1
0.15
Figure 4.2. The blue line represents the first measurement and the red line the repeated
measurement. The top inset (a) shows the early part of the seismic record and the bottom
inset (c) shows a time window of the late part (coda). As described in the text, the whole
waveform is highly reproducible.
58Chapter 4. Monitoring stress change in an underground mining environment with coda wave interferometry
the coda of two repeated measurements match wiggle by wiggle (figure 4.2c). Geophysicist
often discard coda measurements because of their complexity. Their high repeatability
shows that they are different from noise and carry information about the medium they
propagated through. In the late coda, the noise level in the un-stacked data (vibration
noise, instrument noise, air movement) is ten times smaller than the signal amplitude. We
estimate the noise of the seismic records from that part of the measurement before the first
arriving wave. Since in that section of the record no waves have yet reached the receiver,
we only measure noise.
In order to quantify the reproducibility of the seismic waveforms we calculate a timeshifted correlation coefficient Rmax (maximum of equation (4.2)) of 0.96 for two repeated
measurements, which confirms the high repeatability of seismic coda measurements in the
underground mining environment. In other words, if all the recording parameters (source,
receiver, instrumentation) stay constant and nothing changes in the rock-mass, we can
reproduce the same seismic measurement to a high degree of precision. If there is a change
in the rock (e.g. stress, modulus of deformation) the seismic measurements will change too.
If we apply coda wave interferometry to our data, then the velocity perturbation that
we infer is averaged over the entire pillar as well as over shear and compressional waves.
However, even a cursory examination of the coda shows that after the transient has died out
we have a nearly monochromatic (constant frequency) signal. The frequency measured is
about 500 Hz. This frequency is consistent with the fundamental (lowest frequency) mode
of a surface wave bouncing up and down between the base and roof of the pillar (about 3 m)
along the surface near the source and receiver. It is also consistent with a higher overtone
(or harmonic) of a body wave propagating back and forth within the pillar (25 m across).
The measurements could be explained by several modes that match with the geometry
of the pillar, velocity and frequency of the waves. We would need further information to
conclusively determine what the type of wave mode is recorded. For example, we can use
an array of receivers and compute the direction of wave propagation (Scales & Malcolm,
2003).
4.6
Decorrelation of coda waves with stress change
If we increase the pressure in the cell, we change the seismic velocity and thus the
recorded waveforms are different. When the change in the medium is small, we only change
the coda waves and the early arriving waves will remain unchanged. Figure 4.3a shows two
waveforms, one recorded at 600 psi and one at 1800 psi. The early arriving waves (figure
4.3b) are invariant to the stress change and cannot be used to monitor a stress difference
of 1200 psi in the slot. Consequently, tomographic images based on one-way travel-times,
before and after the stress change, would be identical and cannot be used to monitor the
stress difference. The coda waves, however, show a decorrelation of the waveforms (figure
4.3c). Hence, due to their increased sensitivity, coda waves can be used to monitor the
velocity change that is induced by a stress difference of 1200 psi.
The velocity change in the medium does not lead to an erratic decorrelation of the
coda but rather a coherent time lag or phase shift (figure 4.3c). We compute this time lag
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Time-Lapse Monitoring with Coda Wave Interferometry
0.02
59
0.01
0
-0.01
0.01
-0.001
0
0.001
0.002
0.003
0.087
0.088
0.089
0.09
0
-0.01
0.001
0
-0.02
-0.001
0.086
-0.03
0
0.05
Time (s)
0.1
0.15
Figure 4.3. Two waveforms, one measured at 600 psi of pressure in the cell (blue) and one
measured at 1800 psi (red). The top inset (a) shows the the early time window and the
bottom inset (c) the coda.
60Chapter 4. Monitoring stress change in an underground mining environment with coda wave interferometry
δt
0.9
0.6
R
0.3
0
-0.3
-0.6
-0.9
-0.01
-0.005
0
time lag (s)
0.005
0.01
Figure 4.4. Cross correlation function of the data shown in figure 4.3c. Note that the
maximum of the cross correlation function is centered at the time lag δt, which is the value
we estimate from the coda.
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Time-Lapse Monitoring with Coda Wave Interferometry
61
0.03
0.015
0
-0.015
-0.03
0
0.05
0.1
0
0.05
0.1
time lag (s)
0.003
0.002
0.001
0
time (s)
Figure 4.5. The top figure shows the seismic record. The bottom figure shows different
estimates of δt for multiple time-windows, therefore providing a consistency check. The
first three points are computed from the early part of the waveform and are not sensitive
enough to detect the introduced velocity change.
62Chapter 4. Monitoring stress change in an underground mining environment with coda wave interferometry
0.004
0.0035
0.003
dv/v
0.0025
0.002
0.0015
0.001
0.0005
0
0.06
0.08
time (s)
0.1
0.12
Figure 4.6. Velocity change estimates for eight non-overlapping windows with different
center times. The mean velocity change is 0.25 % and the standard deviation is 0.02 %.
Alexandre A. Grêt
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Time-Lapse Monitoring with Coda Wave Interferometry
63
δt by the lag of the maximum of the cross-correlation function given by equation 4.2. The
cross-correlation function for the coda measurements in Figure 4.3c is shown in Figure 4.4.
By computing δt for multiple, non-overlapping, time windows we obtain increasing values
for δt for increasing total travel-times t (Figure 4.4). These are independent estimates of
travel-time perturbations for given total travel times.
Because of the low sensitivity of early arriving waves, δt is for the early arriving waves
smaller than the sampling interval of the seismic record. This leads to large uncertainties in
the estimated velocity changes (equation 4.1). For that reason, we don’t estimate velocity
changes from the early part of the signal (δt smaller than sampling interval). Furthermore,
we assume a constant velocity change over the whole area sampled by the waves. If that
assumption holds, then the time lag δt grows linearly with total travel-time t (Grêt et al.,
2004c). If the velocity change is localized, however, then there is a non-linear dependency of
δt on t (Pacheco, 2003). In our experiment we introduce a localized stress change around the
pressure cell and would therefore expect a non-linear dependency of δt on t. Nonetheless, we
measure a linear increase of δt with time t (figure 4.5). In order to explain this contradiction,
we speculate that the wave energy that leaks out of the pillar does not come back. Hence,
the waves measured for the entire duration of recording are trapped in the pillar and sample
only some portion of the rock mass in the pillar. In other words, the induced stress change
is locally confined but so is the area sampled by the recorded waves.
We compute the relative velocity change in the rock δv/v from equation (4.1). As
previously explained, we obtain the phase shift δt from the cross-correlation function and the
total travel-time from the time-location of the data window. Since we use non-overlapping
time windows, we get multiple, independent estimates of δv/v (figure 4.6), that provides
a mean relative velocity change and standard deviation. Note that we are measuring a
very small relative velocity change of the order of 0.2 %. This is a value which would be
very hard or even impossible to detect with a travel-time tomography, even with the most
accurate instrumentation.
4.7
Monitoring stress change
In order to test the accuracy of the method we compute the relative velocity change for
four pressure differences of equal amount (1200 psi) but with different starting pressures (0,
600, 800 and 1000 psi, see table 4.7). The measured velocity changes for a stress difference
of 1200 psi each, agree well with the exception of the initial stress increase from 0 psi.
The large velocity increase with the initial stress change can be explained by the closure
of void space, e.g. pores and cracks (Hubbert & Willis, 1957; Friedel et al., 1995). During
the installation of the slit, probably new cracks where formed and old ones opened. These
cracks are closed back by increasing the pressure in the cell from 0 to 600 psi, which is
approximately the pressure from the overburden and corresponds to the stress state before
the installation of the slit. The velocity increase is greatest with an early increment increase
in stress because of this change, voids are easily closed.
The relative velocity change, due to a pressure increase of 1200 psi, measured with
coda wave interferometry is consistently lower than what we would expect from textbook
64Chapter 4. Monitoring stress change in an underground mining environment with coda wave interferometry
Figure 4.7. Table of measured relative velocity change for 1200 psi pressure difference in
the cell and four different starting pressures. The standard deviation is calculated from the
eight non-overlapping windows
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Time-Lapse Monitoring with Coda Wave Interferometry
65
values of about 0.4 % (Carmichael, 1982). In the analysis we assume that the stress change
occurs in the entire pillar, which is not the case in this experiment. What we measure with
coda wave interferometry is a mean velocity change over the full pillar area. This results in
a lower value of velocity change, since some areas of the pillar are not affected by the stress
change. An increase of 1200 psi in the entire pillar would result in a constant velocity change
in the whole area sampled by the coda waves. In that case, coda wave interferometry would
measure a larger average velocity change. In other words, coda wave interferometry would
be even more sensitive to a stress change in the entire pillar and could monitor smaller
stress changes than 1200 psi.
4.8
Discussion
The pressure cell changes the stress in a limited area close to the slot. We can not
tell from the coda waves alone where the stress change is located. In practice, one could
complement coda wave interferometry with local measurements of stress change or deformation. Commonly used instruments are strain gages (Hawkes & Bailey, 1973; Hanna, 1985),
extensometers (Joass, 1993; Sheppard & Murie, 1992) tiltmeters (Joass, 1993) and one-way
travel-time seismic tomography. Seismic tomography in particular lends itselfs to a natural
combination with coda wave interferometry, since the instrumentation can be shared by the
two methods, where tomography gives an image of the stress in the pillar (provided we can
meeasure the change in first arrivals) and coda wave interferometry provides the sensitivity
to monitor small stress changes.
We found that the smallest pressure difference that can reliably be detected in our
experiment with coda waves is 1200 psi. We introduced a localized stress change. When
the same 1200 psi stress change is distributed over the entire pillar, the velocity would
change in the whole area sampled by the coda waves. In that case it is possible to measure
smaller stress differences.
In this study we do not distinguish between S-, P- and surface waves, but an extension
of coda wave interferometry for elastic waves is given by Snieder (2002) who shows that
coda wave interferometry is much more sensitive to S-wave than to P-wave velocity change.
In particular, for a Poisson medium the coda wave interferometry velocity change is given
by
δvP
δvS
δv
≈ 0.09
+ 0.91
,
v
vP
vS
(4.3)
where vS and vP are S- and P-wave velocities and δvS and δvP are the velocity changes for
S and P-waves, respectively.
4.9
Conclusions
We demonstrate that coda wave interferometry is a sensitive tool to monitor stress
changes in an underground mining environment. It is a suitable method when classical
tomography is not sensitive enough to measure a stress change. In addition, coda wave
66Chapter 4. Monitoring stress change in an underground mining environment with coda wave interferometry
interferometry needs only one receiver (geophone) and a hammer source, this is a fraction of
the hardware requirement for travel-time tomography. This makes coda wave interferometry
inexpensive and easy to install. The data processing is basic and can easily be built in a
simple, stand-alone instrument. An instrument can be developed that is a fully automated
monitoring system and resists the rough mining environment.
The results of our study are sufficiently encouraging to repeat the experiments in
an operating mine. The method has potential for monitoring rates of stress change and
deformation in the rock mass surrounding excavations both in underground and surface
mines.
4.10
Acknowledgments
We are thankful to Dave Mosch, who is making the mine a safe place for us to work.
Doug Scott and his collegues for installing the pressure cell in the pillar. Thanks to Matt
Haney, Brian Zadler and Douglas Klepacki who helped preparing the experiment and swinging the hammer. We are also greatfull to Mike Batzle for his continuous help with instruments, pumps and everything else needed to make the experiment work. Thanks to John
Scales for the stimulating discussions about wave propagation. This work was supported by
the NSF (EAR-0106668). We also thank the members of the Center for Wave Phenomena
and the Physical Acoustics Laboratory for stimulating discussions.
Alexandre A. Grêt
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Time-Lapse Monitoring with Coda Wave Interferometry
67
Chapter 5
Monitoring rapid temporal change in a Volcano
with Coda Wave Interferometry
5.1
Abstract
Multiply scattered waves dominate the late part of the seismic coda. Small changes
in a medium, which have no detectable influence on the first arrivals, are amplified by the
multiple scattering and may be seen readily in the coda. We exploit this idea using Coda
Wave Interferometry to monitor temporal changes in the subsurface of the Mt. Erebus
Volcano, Antarctica. Mt. Erebus is one of the few volcanoes known to have a convecting
lava lake. The convection provides a repeating seismic source producing seismic energy that
propagates through the strongly scattering geology in the volcano. Over a time period of
two months, the first arrivals of the seismic waves are highly reproducible. For the first
month this is also the case for the coda. At a certain moment, the seismic coda decorrelates
rapidly. This indicates a rapid change in the subsurface of the volcano, a change that could
not be detected by means of single scattered seismic waves.
5.2
Introduction
The coda of a waveform consists of that part of the signal after the directly arriving
phases (Aki, 1969; Aki & Chouet, 1975). At late times the coda is dominated by multiply
scattered waves. Geophysical applications based on coda waves include earthquake prediction (Aki, 1985; Sato, 1986), earthquake-magnitude estimation (Lee et al., 1972), volcano
monitoring (Aki & Ferrazzini, 2000; Fehler et al., 1998) and monitoring of temporal changes
in the subsurface (Robinson, 1987; Chouet, 1979; Revenaugh, 1995; Poupinet et al., 1984).
Laboratory applications include Diffusive Wave Spectroscopy (Cowan et al., 2002), reversed
time imaging (Fink, 1997), and medical imaging (Li et al., 1997).
Small changes in a medium, which would have no detectable influence on the first
arrivals, are amplified by the multiple scattering and may be seen readily in the coda. We
have previously exploited ultrasonic coda waves to study non-linear temperature dependence
of velocity in granite (Snieder et al., 2002). This non-linearity is related to acoustic emissions
during thermal cracking (Fredrich & Wong, 1986). In contrast to other methods which use
multiply scattered energy, the phase information of the coda is a central part of our analysis.
There are many other possible applications of Coda Wave Interferometry in geophysics,
including dam monitoring, time-lapse reservoir characterization, and rock physics.
68Chapter 5. Monitoring rapid temporal change in a Volcano with Coda Wave Interferometry
Figure 5.1. Mt. Erebus, Ross Island, Antarctica, is currently the most active volcano in
Antarctica. The summit of Mt. Erebus contains a persistent convecting lava lake which
undergoes several strombolian style eruptions daily. Within the past year (2003), small ash
eruptions and a small lava flow have also been observed coming from vents near the lava
lake. Aerial view of the summit plateau and volcanic plume of Mt. Erebus.
The subsurface in the regions of volcanoes is highly inhomogeneous. Such highly
scattering media are attractive for the study of multiply scattering of seismic waves (Wegler
& Luehr, 2001). In this paper we show the application of Coda Wave Interferometry for
monitoring changes in the subsurface of Mt. Erebus, Antarctica.
5.3
Mount Erebus, it’s eruptions and the seismic network
Mt. Erebus, Ross Island, Antarctica, is currently the most active volcano in Antarctica.
The summit of Mt. Erebus contains a persistent convecting lava lake that undergoes several
strombolian style eruptions daily (Rowe et al., 1998). Within the past year, small ash
eruptions and even a small lava flow have been observed coming from vents near the lava
lake (See Figure 5.1 for a photo).
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Time-Lapse Monitoring with Coda Wave Interferometry
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Figure 5.2. Location of the permanent network of nine, single-component short-period
(1-Hz) seismometers and one three-component, 1-Hz station. A infrasonic microphone is
co-located with the summit seismic station.
70Chapter 5. Monitoring rapid temporal change in a Volcano with Coda Wave Interferometry
0
100
200
300
400
500
600
700
800
−1
0
1
2
3
4
5
6
7
Figure 5.3. Five different events recorded at the broadband seismometer station E1S. Event
one occurred on Dec. 12th, event two on Dec. 13th, event three on Dec. 14th and events
four and five on Dec. 15th. Note how well the waveforms are reproducible over a time
frame of days.
The mountain is currently instrumented with a permanent network of nine, singlecomponent short-period (1-Hz) seismometers and one three-component, 1-Hz station, as
well as an infrasonic microphone co-located with the summit seismic station (See Figure
5.2). In 1999, a permanent broadband seismometer was installed along with a tilt-meter and
wind speed and direction instrumentation at station E1S. Stations are powered by gel-cell
batteries recharged by solar panels; many short-period stations now operate throughout the
Austral winter, thanks to sufficient battery capacity combined with their low power consumption (approximately one Watt) (See (Aster et al., 2003) for a more detailed description
of the instrumentation at Mt. Erebus).
Alexandre A. Grêt
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early part
3
4
13
5
14
6
coda
15
16
time (s)
7
8
17
9
18
Figure 5.4. The top of the figure shows the early part of event one on Dec. 12th (red line)
and event two on Dec. 13th (blue line), plotted on top of each other. The bottom of the
figure shows the later part of the same events.
72Chapter 5. Monitoring rapid temporal change in a Volcano with Coda Wave Interferometry
5.4
Reproducible seismic events at Mount Erebus
Figure 5.3 shows five different events recorded at the broadband seismometer station
E1S. Event one occurred on Dec. 12, event two on Dec. 13, event three on Dec. 14 and
events four and five on Dec. 15, 1999. Note the reproducibility of the waveforms over a time
frame of several days. Even when looking at short time windows of the seismic data, the
events are virtually identical (Figure 5.4). Not only are the first arrivals reproducible, the
seismic codas of the two events overlap equally well. This means that during the period when
those earthquakes were recorded, the source, receiver, and subsurface remained invariant.
If either of the three had changed, the seismic waveforms would have changed too.
5.5
Decorrelation of coda waves
Figure 5.5 shows the same comparison of two seismic events, except that the events
occurred two weeks apart in time. The early parts still correlate extremely well but there
is a larger difference in the late parts of the waveforms. Since the source signature (early
window) is reproducible, we assume that the seismic source, a strombolian eruption, for
the two events and the seismometer remained invariant. Since the codas for the two events
are different, however, the medium through which the waves have traveled through must
have altered. We use this difference to monitor a change in the volcano. That change is
too small to have an effect on the early part of the waveforms. In order to quantify the
difference in waveforms, we compute the maximum of the cross-correlation function for the
early parts and the late parts, respectively (Figure 5.6). In the top part of figure 5.6, the
correlations for the early parts (source signature) are high (around 0.9) over the whole two
month period. For the later part of the waveforms, however, the correlations drop sharply
around the 8th of January 2000. This means, that around that date something has changed
in the volcano that can’t be monitored with the early part of the waveforms.
5.6
Measurements at different locations
We have other seismic stations on Mt. Erebus that also show this change in coda
waves. This excludes nonlinear artifacts of the instrument at the station E1S. We processed
four more stations in the same manner as previously described. We find a similar jump in
correlation coefficients at all four stations (ABB, CON, HEL and HOO)(Figure 5.7 and 5.8).
The correlation-level is in general lower than at station E1S because of the higher noise level
and the difference in instrument type (some measurements are clipped). Stations ABB,
CON, HEL and HOO are equipped with short-period (1-Hz) seismometers with a lower
dynamic range than the broadband seismometer at station E1S.
It is difficult to determine what is the cause of the change in the subsurface because of
the complex paths that multiply scattered waves travel. However, there are some scenarios
we can clearly exclude. The decorrelation of the coda waves can’t be due to a change in
seismic velocity only since that would lead to a linear increase in phase-shift with increasing
traveltime (Snieder et al., 2002). We can also exclude all gradual changes with time since
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73
early part
3
4
5
12
13
14
6
7
8
9
15
16
17
18
coda
time (s)
Figure 5.5. A plot similar to the previous one but the two events here occurred two weeks
apart. The early parts still correlate extremely well but there is a larger difference in the
late parts of the waveforms. We use this difference to monitor a change in the volcano.
That change is too small to have an effect on the early part of the waveforms. In order to
quantify the difference in waveforms, we compute the correlation coefficients for the early
parts and the late parts, respectively.
74Chapter 5. Monitoring rapid temporal change in a Volcano with Coda Wave Interferometry
xcorr
1
0.5
0
01−Dec−1999
1
xcorr
early part
01−Jan−2000
01−Feb−2000
01−Jan−2000
01−Feb−2000
coda
0.5
0
01−Dec−1999
Figure 5.6. In the top of the figure the correlation coefficients (all events are correlated
with event number one) for the early parts stay high (around 0.9) over the whole two
month period. For the later part of the waveforms, however, the correlation coefficients
have a sudden drop around the 8th of January 2000. This means, that around January 8
something has changed in the volcano that can’t be seen in the early part of the waveforms.
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Station ABB
0.8
0.6
0.4
0.2
0
01−Dec−1999
1
01−Jan−2000
01−Feb−2000
Station CON
0.8
0.6
0.4
0.2
0
01−Dec−1999
01−Jan−2000
01−Feb−2000
Figure 5.7. In order to exclude nonlinear artifacts of the instrument at station E1S, we
processed four other stations in the same manner. We find a similar jump in correlation
coefficients at all four stations (ABB, CON shown here). The correlation level is in general
lower than at station E1S because of the higher noise level and the difference in instrument
type (some measurements are clipped). Station ABB, CON, HEL and HOO are equipped
with short-period (1-Hz) seismometers.
76Chapter 5. Monitoring rapid temporal change in a Volcano with Coda Wave Interferometry
1
Station HEL
0.8
0.6
0.4
0.2
0
01−Dec−1999
1
01−Jan−2000
01−Feb−2000
Station HOO
0.8
0.6
0.4
0.2
0
01−Dec−1999
01−Jan−2000
01−Feb−2000
Figure 5.8. Time shifted orrelation coefficient of coda waves (in respect to the first event)
for stations HEL and HOO, showing the same rapid drop in correlation as the previous
stations.
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the time of change can be restricted to a few days around January, 8, 2000. This leaves
sudden events such as landslides, movement of lava and associated thermal change, sudden
fluid migration, or slips on faults.
5.7
Conclusions
The strombolian style eruptions at Mt. Erebus provide a repeatable source for monitoring the volcano with seismic methods. The geology of volcanoes is known to be highly
scattering. The seismic waveforms recorded around the volcano are highly reproducible,
not only their early arriving phases but also the seismic coda. The source signature remains
virtually the same over the whole two-month period. The coda, however, has a distinct drop
in correlation around January 8, which suggests a change in the subsurface of the volcano.
The change is not gradual but occurs within a time frame of a few days. Furthermore, the
change is so small that it has no effect on the first arrivals but it’s influence can be seen in
the coda because of its higher sensitivity to a change in the medium.
5.8
Acknowledgments
This work was supported by the NSF (EAR-0106668). We also thank the members
of the Center for Wave Phenomena and the Physical Acoustics Laboratory for stimulating
discussions.
78Chapter 5. Monitoring rapid temporal change in a Volcano with Coda Wave Interferometry
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Chapter 6
Time versus frequency resolution of small changes
6.1
Abstract
Some experiments are most conveniently thought of as being performed in the timedomain, some in the frequency-domain, and some use a hybrid approach. Does the domain
make any difference in the ultimate resolution of the experiment? Or can the details of
the experiment always be tweaked so that the different approaches give the same underlying answer? We consider a toy experiment from optics which lends itself to many such
arrangements. Using both time and frequency measurements, and taking into account the
influence of noise, finite dynamic range, attenuation, and the duration of the measurement,
we address the question for each formulation: how small a change in the system can we
make and still detect it?
6.2
Overview
In many cases, the same basic experiment can be performed in the time or frequency
domain–or some hybrid. An example is spectroscopy. At one extreme, a narow-band,
frequency tunable source such as a laser or microwave oscillator can be used to probe a
system and directly measure the system’s transfer function (Migliori & Sarrao, 1997). In
other situations it might be more convenient to drive the system with a broad-band signal
such as a pulse and look at the impulse-response, as in cavity ring-down spectroscopy
(Hodges et al., 2004) or coda wave interferometry (Snieder et al., 2002). Although, loosely
speaking the methods are equivalent by virtue of the invertability of Fourier transform, in
fact the experimental issues can be very different in these two extreme cases; for instance,
what is the difference between pulsing a system many times and averaging versus driving
the system at a fixed frequencies for a long time and then sweeping the frequency? It is by
no means obvious that the sensitivity of the time or frequency domain measurement of a
particular quantity is the same.
We analyze these issues in the context of a particular experiment. Consider two plane
parallel reflectors as in a Fabrey-Perot etalon (Figure 6.1). Let L denote the distance
between the mirrors and N the index of refraction of the medium between the mirrors;
i.e., the cavity. We consider both wavepackets and monochromatic plane waves normally
incident on the etalon, and we treat the etalon as being large in the vertical and out-ofplane directions, so we only deal with propagation along the x-axis. The latter case gives
the well-known result from optics, although in the analysis that follows we look at both the
80
Chapter 6. Time versus frequency resolution of small changes
fields and the intensities since the same experiment can be performed with, for example,
ultrasonic transducers or microwave transceivers, where the amplitude and phase of the
field can be measured. To make the situation realistic we consider finite measurements
involving finite dynamic range and non-zero ambient noise levels. Further, the index could
be complex, allowing for attenuation.
Ultimately our goal is to consider the following problem. Suppose that a small change
δN is made in the index of refraction between the two mirrors. If the unperturbed index is
Nunp , then the perturbed index is Npert = Nunp + δN . How big does δN have to be before
it can be detected? We treat three domains for the experiment:
• time Launch a (Gaussian) wavepacket into the etalon and measure the field or intensity up to a certain finite time Tmax . All data analysis is done in the time domain.
For instance, we could look at the cross correlation of the signals at T max before and
after the change δN as in coda-wave interferometry.
• frequency Launch a monochromatic plane wave into the etalon; a tunable laser could
be used, for instance, or an ultrasound transducer connected to a function generator.
In any case, we suppose that we can control the frequency and use some sort of
synchronous detection to make the measurement, such as a lock-in amplifier or a
spectrometer.
• time-frequency This will be the same as the time domain above, but instead of
analyzing the data in the time domain we Fourier transform and look at, for example,
the power spectrum, as the etalon rings down. This allows us to incorporate the
interference of the pulses by integrating over all time.
6.3
The Fabry-Perot etalon
The Fabry-Perot etalon is discussed in virtually every optics text; for example, see
(Born & Wolf, 199) and (Lauterborn & Kurz, 2003). Our goal is to find the transmitted
intensity of the system as a function of the reflectivity of the mirrors r, the index of refraction
N , and frequency of the incoming light ν. We refer to Figure 6.1 throughout the derivation.
6.3.1
Time versus frequency in the ideal case
On one’s first pass through this problem, the connections between the time and frequency domain analysis are subtle, so we go through the problem in some detail. The
simplest place to start is with an idealized wave-packet consisting of an infinitely narrow
impulse (delta function) incident on the etalon from the left. That is, the incident traveling
pulse Ei has the form:
Ei = E0 δ(ωτ − kx).
We will use τ for time to avoid confusion with the transmission coefficient.
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81
The reflected field is defined as Er , the transmitted wave field as Et . If there is no loss
upon reflection and transmission at the mirrors, we have a conservation of energy, which
means r2 + t2 = 1, where r = Er /Ei and t = Et /Ei are the reflection and transmission
coefficients of the isolated mirrors.
When the pulse hits the etalon it is split up into an infinite number of transmitted
and reflected pulses. Each transmitted pulse is diminished by a factor of t 2 , where t is the
reflection coefficient of each (identical) mirror, and another factor of r 2m , r is the reflection
coefficient of each mirror and m is the number of bounces inside the etalon. The geometry
is illustrated in Figure 6.1. In addition to the amplitude loss, each bounce inside the etalon
results in time delay of 2N L/c0 , where N is the index inside the cavity, L is the distance
between the mirrors and c0 is the speed of light.
Since we are going to fix our detector position x, we only treat the associated time
shifts. The time τ = 0 corresponds to the time at which we launch our pulse into the etalon
so that the first transmitted pulse arrives at the detector a time τ = N L/c 0 later. Hence
the transmitted field is given by
3N L
NL
) + r 2 (E0 t2 )δ(τ −
)
c0
c0
5N L
+r 4 (E0 t2 )δ(τ −
) + ···
c0
2(m − 1)LN
) + ···
+r 2(m−1) (E0 t2 )δ(τ −
c0
Et (τ ) = (E0 t2 )δ(τ −
(6.1)
Even though we are doing a time-domain experiment (an incident pulse), a great
simplification of this result occurs if we Fourier transform it. So, let us multiply this
expression for Et by e−iωτ and integrate:
Z ∞
Et (ω) =
e−iωτ Et (τ ).
−∞
Each δ-function in the transmitted field will Fourier-transform to an e iωτ term with appropriate delay. The expression for the m-th term is:
r 2(m−1) (E0 t2 )e
−iω
2(m−1)LN
c0
.
So we have the frequency-domain representation of the transmitted field of an impulse
Et (ω) = (E0 t2 )e
iω Nc L
0
+ r 2 (E0 t2 )e
+r 4 (E0 t2 )e
+r 2(m−1) (E0 t2 )e
L
iω 5N
c
0
2(m−1)LN
iω
c0
L
iω 3N
c
0
+ ···
+ ···
(6.2)
82
Chapter 6. Time versus frequency resolution of small changes
Et = E 1 + E 2 + E 3 + · · · + E m
Ei = Eo δt
E1 = (E0 τ 2 )δ(t −
NL
c )
E2 = r 2 (E0 τ 2 )δ(t −
M1
Time τ
N
L 3N L
c )
E3 = r 4 (E0 τ 2 )δ(t −
M2
E1
N 2L
c0
Em = r 2(m−1) (E0 τ 2 )δ(t −
E2
E3
5N L
c )
(2m−1)N L
)
c
Time τ
Et = E 1 + E 2 + E 3 + · · · + E m
= E0 t2 eikL Σr 2(m−1) ei2(m−1)kL
Em
E1 = (Eo t2 eikL )
E2 = r 2 (Eo t2 eikL )ei2kL
E3 = r 4 (Eo t2 eikL )ei4kL
Em = r 2(m−1) (Eo t2 eikL )ei2(m−1)kL
Time τ
Wavenumber k
Figure 6.1. The canonical experiment. A plane wave or Gaussian wavepacket (left) is
incident into an etalon (center ) of spacing L from the left. Each mirror has the same
properties (reflection/transmission coefficients) and the space between the mirrors is filled
with a uniform medium of index N . The index may be complex. A train of Gaussian
pulses is transmitted through etalon (right). There is no dispersion in this example, and
the amplitude decays by r 2 with each transmitted pulse.
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Time-Lapse Monitoring with Coda Wave Interferometry
83
We recognize that this is a geometric series of the form
a + ab + ab2 + ab3 + ... + abm−1 =
a − bl
,
1−b
(6.3)
where l = abm−1 is the last term in the series, and r 6= 1. For us, a = 1 and b = r 2 e2iωN L/c0 ,
which gives us
ET = E0 t2 eiωN L/c0
1 − r2m e2miωN L/c0
.
1 − r2 e2iωN L/c0
(6.4)
At this point we need to distinguish what type of detection we are using. In the case
of ultrasonics and microwave detectors, it is possible to record the full field E that includes
amplitude and phase information. However, in an optical experiment one cannot directly
measure the phase of the wavefield, all we have is the time averaged intensity over some
measurement duration. The transmitted intensity for the Fabry-Perot experiment is:
IT = ET ET∗ = E0 E0∗ t4
(1 + r 4m − 2r 2m cos(2mN Lω/c0 ))2
(1 − r2 eiδ )(1 − r2 e−iδ )
(6.5)
where δ = 2ωN L/c0 , the phase shift for one round trip through the etalon.
We can rewrite Equation 6.5 as
It =
E0 E0∗ (1 − r 2 )2 (1 + r 4m − 2r 2m cos(2mN Lω/c0 ))2
1 + r 4 − 2r 2 cos(δ)
(6.6)
T =
(1 + r 4m − 2r 2m cos(2mN Lω/c0 ))2
It
= (1 − r 2 )2
.
Ii
1 + r 4 − 2r 2 cos(δ)
(6.7)
or, normalizing,
In the limit that the measurement time T M AX → ∞, there are an infinite number of
reflections and since r < 1, we are left with
T =
(1 − r 2 )2
It
=
.
Ii
1 + r 4 − 2r 2 cos(δ)
(6.8)
In the final simplification, we use the identity cos(θ) = 1 − 2 sin 2 (θ/2) to obtain the wellknown form
T =
1
It
=
2
Ii
1 + K sin (N Lω/c0 )
(6.9)
where K = 2r/(1 − r)2 . Figure 6.2 shows a plot of the transmitted intensity as a function
of normalized wavenumber.
To recap, we launched a pulse into the etalon and got out an infinite train of pulses
scaled by powers of the reflection coefficient. To simplify this infinite sum we Fourier trans-
84
Chapter 6. Time versus frequency resolution of small changes
form to the frequency domain and obtained a geometric series for the transmitted intensity.
In the frequency domain, the transmitted intensity shows an interesting dependence on the
properties of the mirror that might be surprising with no other information than the time
series of pulses emerging from the etalon. Even in the case that the pulses are strictly
non-overlapping in time, the frequency domain plot (Figure 6.2) shows the clear signature
of interference.
Readers familiar with this sort of argument will have already observed that the final
result for the spike-transmission coefficient in the frequency domain is the same as if we
had started with time-harmonic plane waves and stayed in the time domain.
6.3.2
Thinking in frequency from the beginning
Imagine a standing wave between the two mirrors. The lowest frequency such wave
with “clamped” boundary conditions E(x = 0, τ ) = E(x = L, τ ) = 0 (putting the origin of
the x coordinates on the left mirror) is:
E(x, τ ) ∝ sin(mπx/L)eiωτ
But this can also be written as a sum two traveling waves going in opposite directions.
If we adjust the relative phase of the two traveling waves we can make them cancel at the
ends of the cavity, thus giving a proper standing wave. Another way of thinking about this
is that a standing wave is really a traveling wave that bounces back and forth an infinite
number of times.
The intensity is periodic for both L and ω, hence we will see characteristic interference
effects either by sweeping through frequency or changing the cavity length. Just like standing waves on a string, the first mode occurs when λ/2 = LN , the second when 2λ/2 = LN ,
and so on. This is the same as saying that standing-waves occur when kLN is an integer
multiple of π:
mπ = km LN =
2π
LN.
λn
(6.10)
Figure 6.2 shows the intensity as a function of frequency of the optical resonator. This
is where an epiphany may occur, and what makes this apparently trivial problem so interesting. Imagine that we have only one mirror with a reflectivity of 0.999, or 99.9% and that the
reflectivity is independent of frequency over some range (readily do-able). A photo-detector
downstream of the mirror will measure 0.1% of the incident light. However, by adding a
second mirror, the etalon transmits 100% of the incident light at certain frequencies! In
effect, we have created an optical filter. The two defining parameters of every Fabry-Perot
etalon are the free spectral range (FSR) and the finesse. The free spectral range is the
distance, in frequency, between successive transmission maxima. This can be calculated by
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Time-Lapse Monitoring with Coda Wave Interferometry
85
1
I t /I i
0.5
∆ν
0
...
(n−1) π
δν
r=0.3
r=0.9
r=0.995
nπ
(n+1) π
...
kL
Figure 6.2. The transmitted intensity as a function of normalized wavenumber (ω = kc) for
different values of reflectivity. Notice that the transmission is 1 at the eigenfrequencies no
matter what reflectivity, whereas in the time domain T m ax decays as 1/log(r).
using Equation 6.10, and that kn = 2π/λn where λn = c0 /N νn .
kn+1 LN − kn LN = (n + 1)π − nπ = π
(kn+1 − kn )LN =
2πLN
(νn+1 − νn ) =
c0
2πLN
∆ν
c0
(6.11)
(6.12)
Equating the two, we find the free spectral range to be ∆ν = c 0 /2πLN . If we denote the
width of the peaks at half maximum transmission (T = 1/2) as δ ν , then by using these two
definitions, we establish the concept of finesse,
F =
∆ν
πr
,
=
δν
1 − r2
(6.13)
a dimensionless quantity that characterizes the sharpness of the peaks. A higher finesse
means we can resolve more spectral lines within our free spectral range, hence a better
interferometer.
6.4
A canonical experiment
Real measurements always have noise, a finite sampling interval and dynamic range.
They might be different from instrument to instrument but we want to know which ones
influence resolution and how.
86
Chapter 6. Time versus frequency resolution of small changes
In order to avoid the complexity associated with correlated noise, we assume the noise
in our data is independent and white (contains all frequencies). Correlated noise can be
included but makes the calculations harder. In the case of uncorrelated noise we can reduce
the noise level by averaging multiple realizations of the measurement. Let be the rms
(root mean squared) noise level after averaging and o is the rms noise of a single realization,
then
o
= √ ,
n
(6.14)
where n is the number of averaged realizations.
The second limitation we face is the finite dynamic range DR of a particular instrument. The dynamic range is the ratio between the largest and smallest measurable value;
it is commonly expressed in decibels (dB):
Pmax
dB,
(6.15)
DR = 10 log10
Pmin
where Pmax is the maximum recordable power and P min the minimum non-zero recordable
power. If we have an instrument that measures in voltage, then the power P is given by
E 2 /Z, where E is the voltage and Z is the impedance of the instrument. Hence,
DR = 10 log10
Emax 2 /Zmax
Emin 2 /Zmin
dB,
(6.16)
where Emax is the maximum recordable voltage and E min is the minimum recordable voltage. If the impedance is constant for all measurements (Z max = Zmin ), equation (6.16)
becomes
Emax
DR = 20 log10
dB.
(6.17)
Emin
DR is often expressed in number of bits B and the relationship between DR in decibel and
B is
DR = 20 log10 2B dB.
(6.18)
Hence, a 14-bit instrument (that is, an instrument with a 14-bit analog to digital converter)
will have a dynamic range of DR = 84.29dB. If we adjust the amplification such that the
highest signal amplitude is 1 Volt and using equation (6.17) we obtain a smallest recordable
voltage Emin of 6.2 · 10−5 Volts.
If Emin is the smallest measurable increment of our instrument, then we can reduce
to be lower than Emin by averaging n = (o /Emin )2 times. After that we can treat the
noise-free case and Emin will be the limiting factor for the amplitude resolution. If we can’t
average n times, however, replacing E min by in the remainder of the calculation will lead
to the right answer, including noise.
Continuing this example we can see that the last measurable pulse coming out of the
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Time-Lapse Monitoring with Coda Wave Interferometry
87
etalon (Figure 6.1) has to be at least 6.2 · 10 −5 Volts; anything below that will be under the
sensitivity of the 14 bits instrumentation. If we increase the number of bits we will be able
to record for a longer time before the signal dies off. But keep in mind that we also have
to average more realizations to have a lower noise level, which may become impractical.
6.4.1
Time domain
It is much more natural, for us at least, to think of experiments in the time domain.
Peculiar non-causal effects in the frequency domain can lead to apparent paradoxes. In the
time domain we visualize the transmitted wave as having bounced back and forth in the
cavity m times
and the amplitude is
Em = Eo r 2(m−1) t2 eikL ei2(m−1)kL
(6.19)
|Em | = Eo t2 r 2(m−1)
(6.20)
We want to arrange things so that the first pulse leaving the cavity has a peak voltage of 1
Volt, since we are assuming this is the maximum we can record with our instrumentation.
Hence, |Ei | = 1/t2 has to be the amplitude of the incident wave E i .
We can now calculate the number of bounces m the wave makes in the etalon before
it becomes undetectable with our 14 bits instrument, meaning that E min = |Em |. From
equations (6.19), (6.20) and (6.18) we calculate
m=
log(|Em |)
+ 1.
2log(r)
(6.21)
The time Tmax it takes for the wave to bounce m times is
Tmax
LNunp
2Nunp Lm
log(|Em |)
=
=
+2
co
log(r)
co
(6.22)
For a 14 bits instrument and a reflection index r of 0.99 we will be able to measure a
wave that has bounced back and forth 483 times and it takes a time
Tmax
LNunp
log(6.29 · 10−5 )
=
,
)+2
log(r)
co
(6.23)
to do so. After that the amplitude of the wave decays below the sensitivity of the instrument.
We now perturb the medium inside the etalon by δN and the maximum recordable time
becomes
log(|Em |)
L (Nunp + δN )
2δN Lm
=
.
(6.24)
T̃max =
+2
co
log(r)
co
The time difference δt that we want to resolve from the two measurement (before and after
the perturbation) is the difference between T max and T̃max , and from equations 6.23 and
88
Chapter 6. Time versus frequency resolution of small changes
6.24 we obtain
2δN Lm
log(|Em |)
LδN
δt =
=
+2
.
co
log(r)
co
(6.25)
Solving for δN in equation (6.25) and for N unp in equation (6.22) leads to the relative
change in index of refraction
δN
δt
=
.
(6.26)
Nunp
Tmax
This is a telling result. Equation 6.26 lets us compute the theoretical sensitivity for a
given experiment and instrumentation. δt is the smallest possible time shift we can measure
with a given instrumentation. It can also be thought of as the smallest sampling interval
our instrument can measure. Tmax is given by Equation 6.22 and depends on the dynamic
range DR and the reflection coefficient r. Dynamic range and sampling interval are set
by the instrumentation and have nothing to do with the physics of the experiment. We
can always make the instruments more sensitive by increasing the number of bits and/or
the sampling interval, up to the limits of modern technology. The reflection coefficient
r, however, is given by the nature of the experiment and it limits the sensitivity of the
time domain analysis. A reflectivity close to 1 leads to a large maximum recording time
Tmax and makes for a very sensitive analysis of the data (Figure 6.4). A small reflection
coefficient, however, will make our analysis insensitive. In other words, if all the energy is
leaks out of the etalon quickly, we can only record few bounces between the mirrors before
the amplitudes are smaller than the dynamic range.
We only considered a single pulse at time T max . But we can repeat the analysis for
each time the wave bounces back and forth between the mirrors. Hence, we can estimate m
√
independent values of δN/Nunp and therefore reduce the uncertainty by a factor of 1/ m.
Furthermore, in this particular experiment we know the path length that the waves
have traveled, namely L times the number of bounces m. By measuring the time it takes
to traverse that path (Tmax ) we obtain the index of refraction N unp and our measurement
becomes an absolute measurement of the change instead of a relative one.
For band-limited data we can still estimate δt by comparing the maxima of the waves.
In the case we include noise, however, it becomes a little more complicated. A suitable
method to estimate δt is the cross-correlation function, which estimates δt directly. We
show how we can get δt from the cross-correlation on an example of Gaussian wave-packets.
In this case we launch a Gaussian pulse into the etalon and record the many echos
until the maximum time Tmax . If we assume, that we can window the signal around the
last recorded echo, we get the following expression for the unperturbed wavefield
Eunp (t) = |Em |e
−π(t−Tmax )2
σ2
,
(6.27)
and for the perturbed wavefield
Epert (t) = |E|e
−π(t−T̃max )2
σ2
,
(6.28)
where a velocity change δN introduces a time shift δt in the perturbed wavefield and can
Alexandre A. Grêt
/
Time-Lapse Monitoring with Coda Wave Interferometry
89
be extracted by means of the cross-correlation function C(t) of E unp (t) and Epert (t). σ
is defined as the full-width at half-max (FWHM) and determines the band-width in our
example. We calculate C(t) by multiplication in the Fourier domain
∗
C(ω) = Eunp (k)Epert
(k),
(6.29)
where the star denotes the complex conjugate in the Fourier domain. The wavefields in the
Fourier domain are given by
2 σ2
e−i2πωTmax ,
(6.30)
2 σ2
e−i2πωT̃max .
(6.31)
e−i2πω(Tmax −T̃max ) .
(6.32)
Eunp (ω) = |Em |σe−πω
and
Epert (ω) = |Em |σe−πω
Inserting 6.31 and 6.30 into 6.29 leads to
C(ω) = |Em |2 σ 2 e−2πω
2 σ2
Inverse Fourier transform gives
C(t) = |Em |2 σ 2 e
−2π(t−(Tmax −T̃max ))
σ2
.
(6.33)
Hence, the cross-correlation function C(t) is centered at δt = T max − T̃max , which is the
object of the time-domain calculation and can be estimated from the location of the maximum of C(t). Computing δt with the cross-correlation function has the advantage that we
estimate δt directly, this is better than computing two large values (T max and T̃max ) and
subtracting in order to get a small value. Furthermore, the cross-correlation function can
be thought as the impulse response (or Green function) of the perturbation in the medium.
In other words, by convolving Eunp (t) with C(t) we can compute Epert (t).
6.4.2
Frequency domain
Ironically, many frequency oscillators create sinusoids by coupling pulses to a stable
cavity. In any case, the key to making a frequency domain measurement is having a stable,
tunable oscillator. Of course, all measurements are made in laboratory time, but if we
measure any frequency dependent quantity A(ω) at enough frequency points we have, in
effect, measured the Fourier Transform of its time-domain response A(t). A measurement
at a given frequency corresponds to driving the system under test with an long sinusoid and
measuring either the amplitude and phase of the response (or quadrature components, i.e.,
the Cartesian components of the amplitude/phase), or for optical measurements, measuring
the time-averaged intensity of the signal.
This means we cannot think of rays reflecting in time off the mirrors any more. We
must think in terms of modes of oscillation. We want to find the value δN associated
with an index perturbation in the cavity. To do this, we will measure the frequency-shift
associated with any one of the cavity modes.
90
Chapter 6. Time versus frequency resolution of small changes
We have already shown that for a large number of reflections the frequency-dependent
transmission intensity is Equation 6.9
T =
It
1
=
.
2
Ii
1 + K sin (N Lω/c0 )
If we look at Equation 6.9 our first thought is that in order to get a transmitted intensity
(IT ) of one, the argument of the sin must be zero. This is not the case, however, since none
of the parameters (N, L, ω, c0 ) are zero. The argument is any integer multiple of π.
The ideal way to measure this frequency shift is to perform a frequency-sweep of the
unperturbed system and find one of the frequencies of maximum intensity (I T ) as shown in
Figure 6.2. When the system is perturbed by some δN , perform the frequency sweep again
in the same neighborhood to obtain the new frequency of maximum intensity. This gives
δν, the frequency shift.
As previously mentioned, we would like to find the relative sensitivity δN/N unp . We
begin by rewriting kLN as
kLN =
ν
1
2πLN = 2πLN.
λ
c0
Replacing kLN and solving for the index of refraction gives


v
1 2πL
u 1 − T I 
arcsin u
N =±
2 .
t
ν c0
2r
TI
1−r 2
(6.34)
(6.35)
Negative index of refraction materials do exist, but for simplicity we consider only positive
N . The first thing to notice is that ν depends on 1/N , so when N is increased to N unp +δN ,
our observed frequency maximum decreases. We denote the perturbed frequency as ν pert
so that νpert = νunp + δν.
δN = Npert − Nunp =


v
u
1 − TI 
2πL
δν

=
arcsin u
2 
t
2
νunp − νunp δν
c0
2r
TI
1−r 2
(6.36)
δν
δN
=
.
Nunp
ν + δν
(6.37)
Finally, we construct the ratio δN/N to obtain
The sensitivity is dependent upon the frequency shift, much like the time-domain
method is dependent upon a time shift. However, there is an additional dependence on
frequency. For higher frequencies, or higher modes excited in the cavity, the frequencydomain measurement is more sensitive because the same perturbation δN results in a larger
Alexandre A. Grêt
/
Time-Lapse Monitoring with Coda Wave Interferometry
91
frequency shift.
Note that we can use the spacing between adjacent intensity maxima to obtain an
absolute value for the index of refraction N via ν n+1 − νn = 2N L/c0 since we know the
cavity length. That means we can also obtain an absolute measure of the perturbation δN
from Equation 6.37.
The most interesting thing to notice is what happens if we measure the frequency
shift at one of the resonant modes. In Equation 6.7, the denominator becomes unity
and for a sufficiently large number of reflections (m large), the numerator (1 + r 4m −
2r 2m cos(2mN Lω/c0 ))2 approaches unity for all values of reflectivity r. That means our
measurement is independent of the reflectivity of the mirrors in the frequency domain! This
is not the case in the time domain.
6.4.3
Time-frequency domain
In the previous discussion of the time-domain problem a single pulse was analyzed
before and after the perturbation. This was then extended to multiple pulses as a way of
improving the signal-to-noise ratio of the one-pulse estimate of the perturbation. In doing
so, however, we were implicitly treating the pulses as being independent. But the pulses are
not independent since they have a common origin. This is an example of phase coherence.
This coherence manifests itself as an interference effect. Even if the pulses do not overlap
in time, provided they are phase coherent when we integrate over all time (by going to the
Fourier domain, for example) the interference fringes will show up. These fringes are the
key to improving the resolution of the independent-pulse treatment.
The fact that a train of phase-locked pulses gives rise to a harmonic comb of frequencies
is routinely exploited in optics (e.g., (Teets et al., 1977), (Bellini et al., 1997)). The idea is
very simple. Consider a Gaussian pulse measured at a fixed location
2
2
1
e−(τ −∆τ ) /2σ .
E(τ ) = √
2πσ
The time shift introduces a phase shift in the Fourier transform:
r
σ −iω∆τ −σ2 ω2 /2
e
e
.
E(ω) =
2π
Hence, the power spectrum (i.e., the intensity in the frequency domain) for two identical
pulses separated by ∆τ will contain an interference term. The envelope of the power
2 2
spectrum will still be the Gaussian e −σ ω /2 , but there will be modulation with peaks of
full-width at half-max (FWHM) π/∆τ and separation 1/∆τ . For m such pulses the power
spectrum is a geometric series the summation of which is:
σ −σ2 ω2 1 − cos(ωn∆τ )
e
.
(6.38)
Im (ω) =
2π
1 − cos(ω∆τ )
In the m−pulse case the spacing between the peaks of the power spectrum is still 1/∆τ but
92
Chapter 6. Time versus frequency resolution of small changes
the FWHM is 2π/m∆τ . Provided ∆τ is constant, then for large m, the pulse superposition
provides a precise ruler in the frequency domain, allowing pulse trains to achieve bandwidth
far below that of a single pulse (see (Udem & Fergusen, 2002)).
This is almost the result we need. To take into account the nature of the etalon, we
simply have to incorporate a decay which goes as r 2 for each back-and-forth reflection in
the etalon and an overall factor of t 2 for the transmission. Also, it is convenient to shift the
origin of time so that the first pulse through the etalon arrives at the detector at τ = 0.
Then, one can show that the total field after m bounces in the etalon is:
2
2
t2 e−σ ω /2
Em (t) = √
2πσ
"
1 − r 2(m+1) e−iω(m+1)∆τ
1 − r 2 e−iω∆τ
#
(6.39)
This is actually extremely close Equation 6.4, recognizing the time delay ∆τ = L/c.
In fact the only real difference is that in 6.4 the envelope of the transmitted field in the
frequency domain is flat, since the incident field is just e iωt , whereas now we are considering
Gaussian pulses. So when we compute the intensity associated with Equation 6.39 in the
limit of a large number of bounces m we get
E(ω)E(ω)∗ =
1
σ 2 −σ2 ω2
t e
2π
1 − 2r 2 cos(ω∆τ )
(6.40)
For the finite-time case we have the transmitted intensity after m bounces
σ 2 −σ2 ω2
t e
2π
1 + r 4m − 2r 2m cos(ωm∆τ )
.
1 + r 4 − 2r 2 cos(ω∆τ )
I(ω)t =
(6.41)
This result can be seen to be a product of a Gaussian envelope and a trigonometric term
associated with the interference of the multiple pulses. The interference term is identical to
Equation 6.7 when we swap ∆τ for the equivalent N L/c 0 .
The key idea is that we can exploit the fringes of the interference term to enhance the
resolution of the estimates of the perturbation, over what would have been the case with
independent pulses. The only difference between this result and the earlier pure frequency
domain one is that in the frequency domain the incident field is a monochromatic plane
wave and hence the envelope is flat.
For pulses, the breadth of the envelope of the frequency domain intensity is inversely
related to the pulse width; this breadth does not depend at all on the time of the recording
time. Whereas the FWHM of the interference fringes depends on the number of bounces. So
this brings in the finite recording time (and hence the attenuation of the cavity) in a fashion
that is mid-way between the pure time-domain and the pure-frequency domain approaches.
Alexandre A. Grêt
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Time-Lapse Monitoring with Coda Wave Interferometry
93
Output: FREQUENCY
Input: TIME
Input:
Gaussian pulse
bandwidth: finite
r=1
Output:
zero line−width interference with
finite−width Gaussian
envelope
Input:
plane wave
bandwidth: infinite
r=1
Output:
zero line−width interference with
infinite−width Gaussian
envelope
Input:
plane wave
bandwidth: infinite
r<1
Output:
finite line−width interference with
infinite−width Gaussian
envelope
Input:
Gaussian pulse
bandwidth: finite
r<1
Output:
finite line−width interference with
finite−width Gaussian
envelope
Figure 6.3. Left: Time-domain input signal. Center: Descriptions of experimental inputs
and outputs. Right: Frequency-domain output signal (or Fourier Transform of time-domain
output).
94
Chapter 6. Time versus frequency resolution of small changes
6.4.4
Comparison of the different methods
Conceptually an impulse-response (time-domain) experiment is the easiest to visualize,
and usually perform. As a pulse propagates through the medium, the properties of the
medium are impressed on it. If the pulse is somehow confined, then it samples the medium
many times. In the limit of no attenuation or loss, the incident pules samples the medium
indefinitely. If we make a small perturbation to the system, so small that it is not visible
in the first pulse, it may be that if we wait long enough we can see the effect at late times.
This is the basic idea of Coda Wave Interferometry and Cavity Ring-Down Spectroscopy.
At the simplest level, we can simply compare two late-time pulses (before and after the
perturbation). Cross-correlation of these two pulses gives, in effect, the impulse response
of the perturbation. This two-pulse approach can be extended statistically by considering
many pairs of pulses independently.
But we know that the pulses are not independent, they arise from one source so they
are phase locked. As a result, not matter how far separated the pulses are in the time
domain, when we Fourier Transform to the frequency domain, we get interference fringes in
the system response. These interference fringes become increasingly sharp as the number of
pulses in the Fourier Transform increases. Ultimately we can sharpen the fringes no more
than the natural line-width associated with the Finesse of the cavity. And as we showed
in the text, the envelope of this system response function is governed by the bandwidth
of the incident pulse. As the pulse becomes more and more delta-like, the envelope in the
frequency domain becomes more and more flat. In the limit of true delta-function spike,
we recover the frequency domain approach, which corresponds to driving the system with
a pure sinusoid at a fixed frequency and then sweeping the frequency over some range.
The situation is summarized in Figure 6.3. At the top we see an incident Gaussian
pulse. It bounces back and forth many times in the cavity and the Fourier Transform
of the response is a Gaussian envelope with interference fringes. The spacing between the
fringes depends on the cavity properties, while the width of the fringe depends, additionally,
on the number of bounces. If the reflectivity of the mirror approaches 1, then at the
eigenfrequencies of the cavity, we have an infinite number of bounces and the line-width
goes to zero. As a corollary, in between the eigenfrequencies, no light gets through at all.
This is what we call the time-frequency domain.
In the next panel we show the result of applying a fixed sinusoidal input signal. As
we sweep over frequencies we get the same result as in the time-frequency approach except
that the envelope of the fringes is flat, since we’re putting in a constant amplitude at each
frequency.
The bottom two panels show the same results but in the case of a more realistic (r < 1)
mirror. In this case there is a natural linewidth associated with the fringes that depends
on the finesse of the cavity. For the pulse experiment at the bottom, there is an additional
linewidth associated with the number of bounces in the etalon. In the limit that this
becomes large, the linewidth of the time-frequency method approaches the pure frequency
domain method.
In addition to the interference fringes, there is another fundamental, physical difference
/
Time-Lapse Monitoring with Coda Wave Interferometry
95
frequency domain
time domain
sensitivity
Alexandre A. Grêt
reflection coefficient
sensitivity
frequency domain
time domain
frequency
Figure 6.4. Comparison of time- and frequency-domain sensitivities (δN/N ) as a function
of frequency and reflection coefficient. A higher sensitivity corresponds to a lower value on
the vertical scale.
96
Chapter 6. Time versus frequency resolution of small changes
between the time-domain and the frequency-domain approach. In the time domain we can
measure a finite number of bounces between the mirrors depending on the reflectivity. We
can put more energy into the system by averaging many traces, which will decrease the noise
level but not increase the maximum recordable time and the number of bounces. Hence,
the sensitivity of the time-domain method depends on the reflectivity even in the noise free
case (see top part of Figure 6.4 top). In contrast, we can drive the system in the frequency
domain as long as we want, also leading to a noise-free signal where the sensitivity does not
depend on the reflectivity of the mirrors (Figure 6.4 top).
Similarly, the sensitivity of the frequency-domain measurement depends on the particular frequency we use, where the method is more sensitive at higher frequencies. In contrast,
the data analysis in the time-domain has a constant sensitivity independent of frequency
(Figure 6.4 bottom).
6.5
Acknowledgments
We acknowledge many useful discussion with Roel Snieder, Kasper van Wijk and
Martin Smith. This work was partially supported by the National Science Foundation
(EAR-0337379).
Alexandre A. Grêt
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Time-Lapse Monitoring with Coda Wave Interferometry
97
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